'L I B RAHY OF THE UNIVERSITY Of ILLINOIS 510.84 no. 257-264 cop. 2 CENTRAL CIRCULATION AND BOOKSTACKS The person borrowing this material is re- sponsible for its renewal or return before the Latest Date stamped below. You may be charged a minimum fee of $75.00 for each non-returned or lost item. Theft, mutilation, or defacement of library materials can be causes for student disciplinary action. All materials owned by the University of Illinois Library are the property of the State of Illinois and are protected by Article 16B of Illinois Criminal law and Procedure. TO RENEW, CALL (217) 333-8400. University of Illinois Library at Urbana-Champaign JUW 2 li 1UUU When renewing by phone, write new due date kolnnt r\rflir1A1ie /11a /llto 11 AV below previous due date L162 Digitized by the Internet Archive in 2013 http://archive.org/details/onuseoffinitefie263nusp >" Report No. 2 ^3 yyu^C^ ON THE USE OF FINITE FIELD BASED MDDELIN I N FO LYNOMI AL MAN IF J LATI ON by Stephen J . Nuspl Jane 1, 1968 Report No. 263 ON THE USE OF FINITE FIELD BASED MODELING IN POLYNOMIAL MANIPULATION by Stephen J, Nuspl June 1, 1968 Department of Computer Science University of 'Illinois jrbana,, Illinois 6l801 This work was submitted in partial fulfillment of the Doctor of Philosophy in Electrical Engineering and was supported in par 1 : by the National Science Foundation under Grant No „ NSF-GP-4636 „ ON THE USE OF FINITE FIELD BASED MODELING IN POLYNOMIAL MANIPULATION Stephen John Nuspl, Ph D» Department of Electrical Engineering University of Illinois, 1968 A major hindrance to the manipulation of polynomials by a computer is the rapid growth of the polynomial coeffi- cients during the application of some frequently used algor- ithms o The methods presented in this thesis to try to alle- viate this problem and to increase the efficiency of polyno- mial manipulation are based on the use of algebraic redundancy Let Iq be the set of all integers and I« + i d © the adjunction of I. consisting of the set of all polyno- mials in x* +1 with coefficients in loo Let F Q be a field of order p Q where p Q is a prime in I Q o The exten- sion field P, +1 is defined to be the set of residues in F.[x. + ,] modulo p. ,„ a monic irreducible polynomial in ^1^ x 1+l^° ^ he sets I* are defined as the modeled domains and the sets F, and their adjunctions as the model domains* A modeling function which maps any element of a modeled domain into any one of the model domains in defined and shown to be a homomorphisnu Some of the models are capable of retaining the degree information of selected indeterminates in the modeled polynomials o A virtual degree augmentation is added to the model and its algebraic treatment is developed in such a way that it is able to detect when a model polynomial has the same degree as the modeled polynomial without looking at the latter o A major result based on the augmented modeling is the model conclusive g.c.d. theorem which gives the conditions under which two polynomials can be shown relatively prime by performing the g Cod» algorithm only on their models » Since model domains in which the coefficients are finite field elements can be chosen, the computational effort required can in many cases be orders of magnitude less than if the goC.do algorithm were applied directly to the modeled polynomials o Further applications of modeling are discussed but not developed to any depth* ill ACKNOWLEDGMENT The author would like to thank his advisor. Professor Jo E Robertson, for his guidance, encouragement and support given during the development of the ideas in this dissertation and for his suggestions leading to the improvement of its presentation » Thanks are due to Mrs. G c Polizzotti for her patience and meticulous care in the preparation of the manuscript o iv TABLE OF CONTENTS CHAPTER PAGE ACKNOWLEDGMENT . . • . . . . . . . . . . . . . • . . . iii I INTRODUCTION . . . . . c . o . . . . . . . . . • . ° 1 II BACKGROUND , POLYNOMIAL DEFINITIONS , DOMAINS „ . , . 6 2ol Introduction 00 o<..o»ooo»<.oooo . 6 2.2 Notation , Conventions , Polynomial Rings , a . . 6 2o3 Polynomials over Integral Domains , Finite Fields 14 2o^ The Modeled Domains ..» a «.««eeoeo« 17 2. 5 The Model Domains O o»»oo.».«oo oo 20 2o6 Further Conventions on Notation „ o , . » « <» » 27 III THE RECURSIVE MODELING FUNCTIONS . , , . o „ . » . » 29 Jol XnurOClUC U 1 On oooooo.oeooooooooo £y 302 The C Modeling Functions „ » • • o <> . • o • . 30 303 The C Modeling Theorem Proofs » « <,<,<> » 31 304 The D Modeling Functions » » • « . . » e e o o hj 3«5 The D Modeling Theorem Proofs . • „ « » • 44 3<»6 The Recursive Modeling Function and its w omp on en uSoooooooooo°oooooooo ^c. 3«7 Summary of Domain and Modeling Relations „ „ <> o 58 IV AUGMENTED MODELING . . . . , „ „ „ . „ . . „ „ , 61 Toi iriLr oquc 1 1 on 000*00000000000000 01 4o2 The Augmented Modeling Domains and Functions . 62 4.3 The Model Conclusive goCd. Theorem » o . <> e 73 4.4 On the Use of Augmented Modeling in Elimination 83 V THE PRIME CONTEXT SEQUENCE . . „ . „ „ . • » . . » 89 joi in LroQuc ui on 000000000000000000 07 ^o^PQooooooooooooooooooooooo 0^7 .2 _y,P^P-*>*^U o o o o o o o o o o o o o o O O O O O ^^T VI CONCLUDING REMARKS . . . . . . . . . . . . . . . 98 «Vx * IjiNi/1 A, Aooo 00000 000000 00000 o o o o XU-L. ***^ * dXM 1/lA D o o o o o o o o o 9 o o o a o o • o o o o o o X v ^ ixsltr JirirjJ\IC r*o • oooooooo»oo©o»*«oooooo lO*^ VITA in< w * **» 00 000000000 000 00 00 ooe 00 00 o-LW") CHAPTER I INTRODUCTION The increasing use of the computer as an aid in algebraic manipulation makes it desirable to study techniques by which the speed of performing a manipulation is increased and the amount of physical equipment required during the mani- pulation is decreased o Of central importance are techniques by which polynomials in several variables are handled since most problems eventually filter down to a total description by polynomials or a description in which polynomials play a key roleo In most practical cases the polynomials are defined over the integral domain of integers and consequently it is algorithms in these polynomial domains which should be optimized for speed and minimality of equipment o The method proposed in this thesis to accomplish this is to add to the internal representation of the polynomial redundant information in such a way that the redundancy may be used to determine some desired properties of a polynomial without requiring that the entire polynomial be scanned for the property o For example „ we may have two polynomials in a number of variables and we may wish to ask the question "Is polynomial 8 a 9 identical to polynomial °b" in the sense that if both are put into the same canonical form and ordering for every term in "a 1 there is an identical term in 8 b»? 98 The straight forward way of answering this question is to put "a" and "b 8 into the same canonical form and compare them term~by~term until a mismatch occurs or until the terms are exhaust edo When such an algorithm has a very high probability of producing the answer "no" and if the much more concise redundancy is able to state conclusively that "no 11 is the answer, considerable total time can be saved by first trying to answer the question only with the redundancy „ Then the actual polynomials must be used only if the redundancy does not produce a conclusive "no" „ In either case the answer will be the same as when only the actual polynomials are used but because of the high probability of a "no" the comparative average time for answering the question can be orders of magnitude lower when the redundancy is used. Of particular interest is redundant information which retains some of the structure of the polynomials „ When operations are performed on the redundancy and the result gives some information about the effect of applying the equi- valent operations on the actual polynomials, the redundancy can be thought of as being a "model" of the actual polynomial . Since this will be the case in the development of this thesis „ we will define the sets of all polynomials over the integral domain of all integers as the "modeled domains", the sets of all models as the "model domains" and the process of mapping a modeled polynomial into its model as a "modeling function 90 « The modeling used in this thesis is based on finite fields and their adjunctions in the sense that the model domains are either finite fields or adjunctions of finite fields o The usefulness of this particular type of modeling stems primarily from the fact that the modeling functions which will be defined are homomorphisms from the modeled domains into the model domains which produces the desired effect of preserving some of the structure during the modeling o It will be shown that some of the modelings which will be defined have the additional property of being capable of retaining the degree information of selected variables and burying the rest in the finite field coefficients of these variables,, The practical significance of this is that the degree information of the selected variables can be used for decisions in algorithms while at the same time the coefficients are finite field elements which Implies that they are automatically restricted in size Normally the problem in polynomial manipulation is that the coefficients grow so rapidly during the application of an algorithm that it is impractical to work with other than almost trivial polynomials in a few variables even when a very fast computer is used to perform the manipulation,, The only other effort known to the author in which finite field theory was used as an aid in algebraic manipula- tion is ref (8) by W„ Ao Martin On the one hand his approach was much more ambitious than that contained In this thesis since the modeling was extended to functions of a complex variable o However „ his models were restricted to take on values only in a prime field which resulted in the loss of all degree information in the models . It is precisely the pre- servation of this information which gives so much power to the modeling developed here even though it is restricted to poly- nomials over integers o Finite field theory has been found useful in determining the existence of solutions of equations , so it is not too surprising that the same theory can also play an important role in practical manipulative work with polynomials o The development of the modeling is first started in Chapter 2 with the review of some concepts in modern algebra which are later required 9 a rigorous definition of the modeled and model domains and some relations among these domains » The modeling functions are defined and proven homomorphisms in Chapter 3° The detailed homomorphism proofs are included mainly for completeness s they need not be studied in order to understand the modelings Finally, a summary which should help to unify and to make more understandable the relations between the various modeling functions is presented in section 3o7o Chapter 4 introduces the additional concept of a virtual degree augmentation to the models „ A set of operations defined on the augmented domains are developed in sufficient depth to be able to arrive at an adequate set of functions which permit us to define a greatest common divisor (g Cod ) algorithm which operates on elements in the model domains <> It is then shown by the "model conclusive goCodo theorem' 8 that two polynomials can be shown to be relatively prime under specified conditions by using only elements in the model domains during the computation,. The practical signi<= ficance of this is demonstrated with a set of examples „ In Chapter 5 some practical aspects of operations in the prime subfield and in other model domains are briefly considered o The appendix includes an example of a complete set of models for a polynomial in 3 variables and a tabula= tion of a set of primes with selected properties „ CHAPTER II BACKGROUND, POLYNOMIAL DEFINITIONS, DOMAINS 2d Introduction In order to define the polynomial modeling process rigorously it is first necessary to review the definition of a polynomial ring and the operations which are related to the modeling o The required theory Is based on concepts in modern algebra some of which will be stated in section 2o2» Much of the material is very elementary for someone well acquainted with algebra o but its inclusion is necessary in some cases for the explicit referencing of well<°known theorems and In other cases it Is used as a means of defining the notation intrin- sically o For example the definition of a group Includes the normally used existential quantifier (a)„ the universal quantifier (V) and the symbol ( 3 ) standing for "such that"o The reader not familiar with a concept or symbol used should consult one of the references (1), (10) „ (11) „ 2o2 Notation,, Conventions „ Polynomial Rings Let {S O} be a monoid in which S is a set of elements closed under the binary associative operation O: SXS — > S* Let and 1 represent the Identity element In S when the operation O is respectively addi- tive and multiplicative o The sum symbol , Yo will then be defined as follows? n+1 n For a ± € S 2^0 (a i ) ■ a Q g Jo (a ± ) - ( ^O (a^Oa^^ i=0 1=0 1=0 for n > 1 When O Is additive we make the additional convention? for a a = a € S 9 n > define na « }o (a. ) where 1^0 a 4 = a < i When O is multiplicative we define a n - )o (a. ) where < ifro \a 1 = a < i . It is immediately apparent that for m > 9 n > ma O na = (m+n)a when O is additive a m O a n = a m+n when O is multiplicative, {O O} Is a group if G is a non=>empty set < 2o2<>01> and O is a binary operation on the cartesian product GXG with the following properties? 1. O i GXG — > G (Closure) 2* a b © € G s> aO(bOc) ■ (aOb)Oo-(Associatlvlty) 3» 3e€Go9olfa€G c aOeaa ^o va€G 3b€G o B 8 aObse 8 Definition s {R © ©} is a ring if {R,©} is a commutative <2o2o03> gr oup and © is a binary operation on RX R with the properties? lo © § RXR > R 2o a b c e R => a © (b c) » (a © b) © c 3o a b c 6 R ■> a © (b ® e) ■ (a © b) ® (a © c) and (a ® b) © c ■ (a © c) © (b © c) The ring {R,®,©) will be referred to as "the ring R with respect to the operations @ © w or when it is understood that R forms a ring with respect to the operations ® © simply as "the ring R" or "R"o Definitions Let {R „■+„<>}„ {R 10 ® ©} be rings and <2o2o0^> x „ R __ > R Then x ls a rlng nomo „ morphism from R into R, if ^a b 6 R aX © bX = (a -t b)X aX © bX = (a ° b)X Q Under these conditions we denote X by X s R -§-> R x o Definition ? e is called the multiplicative identity or unit of R if e©a^a©e = a for each a in Ro If e is in R 9 R is said to be a ring with an identity o zt = {i I 1 Is an integer and < 1} Definitions An infinite R-sequence is defined by the fol- <2o2o0?5 lowings Let {R @ 9 ©} be a ring with identity, Let 9 1 be the zero and unit in R An R=sequenee is then defined to be a function f s Z Q *=— > R where (i)f ■ t ± with I € zj t ^ 6 H, f is denoted by f ■ (tq D t^ t^ 9 oo e t T s 9 o o o ) o r. is sometimes called the i'th coefficient or 1'th component of f Q Definition ? R^ « [u | u is an infinite Resequence? <2c2o08> u m ( U0o11iOGOO ) ^a s n «3. ? i > n, u t s 0} The last condition is equivalent to saying that only a finite number of the components of u are not the zero element in R c Definition s Let u e v € H™ us (u~,,u, 9 u <2o2o09> R where u|fs Cu Q © b o*- u l ® v l° ® % R^Xr — »> R where u © ▼ = ^ e o» G l» G 2 9 « o ) i and e A s £@ (^ © v i~i^ s 7 ® ^ u j ® v fc^ Theorems {R-,*©,©} is a ring with identity in which the zero <2o2ol0> element i s o = (0,0,0,,,,) and the unit is 1 = (1,0,0 9000/0 Definition s R = {u | u = (u ,0 9 0, . 8 ) s u Q € R u £ R-jJ <2o2oll> Theorem s R c R ± and {lo®*®} = {R 8 @,®} where = denotes <2 2 12> «.o«.o t. r ing isomorphism and with the isomorphism defined by the function X s R — — > R where a € R -> aX - (a,0,0 D ,,J, The isomorphism X will some- times be referred to as an embedding of R in R,<, Definition s Scalar multiplication is defined to be the <2o2ol3> binary operation 9 * RXR 1 — — > R 1 where s> a © u = (a © u QO a © u^ a 6 Ro u 6 R 1 ~> a © u = (a © u ft ,a © u, » o ) » Theorem s The complex {{R ® 9 ©} 9 {R,,®}, ©} is a vector 1? space over R and {x ,x «>x , «, » » } is a set of basis elements when x ■ (0 9 1 . » ) € R-,0 Proofs The vector space properties are readily proven? of 1 interest to us is the fact that {x ,x 0OOO } is a set of basis elements o By our definition of exponentiation, .0 i=0 i=0 = £ © (x) ■ Is x* 3 = V © (x) = (0,0,oo,,l,0,o,o) 11 where the 1 is In the j°th component position of x' For u € R-j^ u = {u 0( u 10 u 2 „ c . » ) ^ e R 9 u can be ex~ pressed as u = ) @ (u* © x )• It follows that 1 {x „x 9 ooo} Is a set of basis elements for R 1 <> Definition s If s € R-. and a © s = s © a fa € R. s ° ° -> is said to be a scalar over Ro Since R is commutative every element in R, is a scalar and of particular importance,, x is a scalar „ Definitions v ^ s said to be algebraic over R if v € R-j and In, a, < i < n such that at least some a^^ + ° and that T @ (a ± © v 1 } = , 1^0 Definition s v Is said to be an indeterminate over R if v 6 R-j, and v is not algebraic over Ro We note that x is an. indeterminate over R Definition. 1 : Let ye E 1 be an Indeterminate over R, 1 n <2o2ol8> Then u = Y ® Cn ± y 1 ) for u i € R 1=1 4- n € Zq is said to be a polynomial In y over Ro The set of all such expressions is denoted by R[y]„ From the definition of R[y] and the previous 1? observation that {x 9 x „x „o 00 } forms a basis for R 1 12 we can conclude that R[x] = R., o Also for some other indeterminate* z, over R we can see that R[z] = R[x] by defining the function X i R[z] — — > R[x] where n n u = V ® (u 1 © z 1 ) £ R[z] and uX = Y ® (u ± $ x i ) X is 1=0~ i=0 easily shown to be an isomorphism from R[z] onto R[x]* Definition ? Let u g R[xL Then it is known that there <2o2cl9> exists an n such that for all 1 > n p u. = where u* is the 1-th coefficient of u Q If u i C u is said to have degree deg(u) = n= If u = or if it is not known whether u = or not u is said to have virtual degree n. The degree of = Y (0 • x ) is defined to be =<* , 1=0 Definition s Let n = deg(u)„ u ^ 0, Then u is called <2o2o20 the leading coefficient of u and^Cu) = u . n Definition ? Let u 6 R[x] Then u is said to be monic <2 2o21> lf/ ( u ) = lo If u v 6 R[x]„ then deg(u ® v) < max(deg(u) c deg(v)) and deg(u © v) < deg(u) + deg(v)o Since R[x] may have divisors of zero the Inequality in the latter is necessary o However „ we will be mostly concerned with integral domains,, in which case we can state 13 deg (u v) - deg(u) + deg(v) N ot^t^on Qonyention: Up to this point the vector operator -^ symbols have been kept separate from the ring operator symbols in formulas. The normal con- vention is to omit or replace the vector and scalar opera- tors with the ring operator symbols or with concatenation for multiplication when no confusion results. Since we will be working with an arbitrarily large number of integral domains and finite fields each with Its own set of operators , we must adopt some of these conventions for compactness in expressions „ but since each of the finite fields will require a different symbol for its operators we must take more than usual care in making these notation conventions o K As examples, let uj ^ R[x3< u = ) © (u, x ) M k=0 r (v @ x m ) c Define N = max(M K) v = > tt) v L, <=, m N Then u ® v = ) # ( (u ® v ) © x n ) -. L> — . n n ° n=0 u © v = y © {( y $ (u. © v j ) © x n ) L„~~ Z^ km n=0 n-k-^m We now make the convention that the scalar product operator symbol © will be omitted and that the vector operator sym* bo3 p will be replaced by the ring operator O, The above 14 K M formulas now reduce to u = \ ® ^ u v x ) » v = / ® ^ v m x ^ k=0 m=0 N u ® v = V ® ( (u n © v n )x n ) n=0 K+M u©v= y® ( y ®(u k ov m ))x n ) n=0 n=k+m In addition we will also sometimes omit the © symbol following a summation symbol when the summation is only a formal representation of a polynomial „ For example three formal representations of the polynomial u would be K K U s= (Ug.UpUp.oo.oo) = \ © (u k X ) = \ U^X k=0 k^O However, the subterm ) ® (u v © v m ) cannot be reduced / 1 k. m n=k+m to V (u^ © v ) since in this particular case the n=kVm symbol V © represents an actual summation using the ring operator © 2o3 Polynomials over Integral Domains,, Finite Fields Definition ; An integral domain is a commutative ring with a unit <2o3,01> , ... and with no divisors of zero. Let (1,+,°] be an integral domain. Then l[x] is the ring of polynomials in the indeterminate x over I. 15 Theorem s {I[x],+,«} is an Integral domain* <2o3o02> Definition s A field is an integral domain in which every aJa J non-zero element has a unique multiplicative inverse* If we define {F 9 ©»0} to be a finite field, then by the above {FCx],©,©} is an integral domain* Definition s Let p 9 a € F[xL p is monic and deg(p) > 1* <2*3°0^> Then by the division algorithm over F[x] there exist unique q,r € F[x] such that a = (q © p) © r where deg(r) < deg(p)* Then define a(mod p) - r* Definition s Let p € F[x]s p is monic and deg(p) > 1* <2*3o05> Define F 1 = (a | a € F[x], degCa) < deg(p)}* Two binary operations on F, are defined as follows 1 1 © s F X X ?! ' — ~> ?2.° a ' b € F l B> a ® b = a © b 1 © s P 1 XP 1 — > F x „ a © b = la b)(mod p) 1 1 Theorems {F 19 9 B ®} is a commutative ring with identity* <2o3*06> Proofs F^ c F[xl* F™ is closed, associative and commuta* -1 1 tive for ® and O and contains the additive inverse of each element in F.,* The distributive property follows from the fact that F, is closed under the operations © X 16 that F[x] is distributive with respects to $ , and that (a © b)(mod p) = a(mod p) © b(mod p) for a„b g F Definition s p € F[x] is said to be irreducible if ^ <2c3o07? a ,,b € F[x] such that p = a b. p is said to be a prime polynomial (prime element or prime) in F[x] if p is a monic irreducible polynomial in F[x] and if deg(p) > 1. 1 1 Theorem s If p is a prime polynomial,, {F, 6 © ©} is a <2.3.08> fleld# Proofs Since p is prime there exist no a b € F, such that a © b = p = (mod p) and hence F, has no divisors of zero. Since F^ is also commutative and has a unit, it is a field Let F be a field and F- c F[x] as defined <2o3.09 in < 2o 3 05> Define the function M by the following s M s F[x] — -> ? 1 where a € F[x] -> aM = a (mod p Th eorem° M is a ring homomorphism from F[x] onto F, ° Je or more concisely M s F[x] onTo > F v 11 2 2 Theorem s Let {R,©,®}, {B,,®,®}. {R 2 »®»®} be rings » Then <2o3o20> ^ i R JL^ R ^ T ^ s R ^ a E_ B> R ^ => T = (TjTg) i R - S -> R 2 Proofs Let O represent either of the operators © or © 1 and let a„b € R« Then (a Q bjTjTg = (a^ O bT x )T 2 *= aT l T 2 ° bT l T 2 usin S the f act that T-, and T 2 are homomorphisms and that aT^bT, € Ri. Hence the composite function T = (T-.T-) is also a ring homomorphism„ 2*k The Modeled Domains In this section we proceed to define a sequence of polynomial rings over an integral domain „ In polyno- mial manipulation on a computer the polynomials with which one is working are normally in theee rings and there- fore we call them the "modeled domains 86 * Definition g The ring's I. , 1 > are defined as follows I Q Is the ring of all integers,, an integral domain with respect to the operations (+„°)o For 1 > 0, let I* -i oe a ring with respect to the operations C+„ v Define 1^ ■ I i „ 1 £x^] By theorem <2o2 o 10> {I lt) + o } is also a ring* The retention of the operators {-** ) is justified by the conventions made in <2c2> Lemmas I* Is an integral domain <2.4.02> 18 Proofs I Q is an integral domain and by <2o3°02> and by in= duction on i, so is I. for < i „ Lemma s I ± c I y < i < j <2^<,03> Proofs Strictly speaking, the statement is not true since ^1+1 ^ s a vec ^ or space over I. e However, an embedding isomorphism can be defined which maps a € I . into j (a 9 9 o o o ) = ax. , € ^-« + -| o The normal convention is to make no distinction between I. and its embedding in I. ,„ a convention we can also adopt since no confusion will result o With this in mind, the lemma holds trivially by induction on h = j-i<, m Definition s Let a 6 I ± ? a = V a fc x i» a m + ° for ° < 1 <2o4o04> k=0 p. oo a = * j - i deg^(a) = V m < j = i max(deg .(a )„ o o o deg.(a m )) < j < i < i < j Lemma s a € I ± => deg.(a) < 0, < i < j. <2o4 o 05> Proofs I. c I. and by the discussion in <2o*K03>< 19 Definition s Define the function D^ , § I ± —> l^ as <2o^ o 06> follows s Let a € I« 8 if I > let m a = I a k x i> a k € X k~0 1-1° aD »1 fa m lk=0 < I < < 1 < 8 =1 where L = {b b = aD,, . „ a 6 1*3 ipj 'i <2o^o0?> Proofs Case Is < i < js by <2o4.03> Case 2s < j < is Let h = i-j Q Assume that — ) a ip X 4 Jul o for some h > Q„ I, cT, Let a 6 I 1 ■i+r a m k~0 -.1 a,.D. a k € I r Then ^ ± ^^ L a k D i,j € X j l € I. since dj the inductive hypothesis and {I^o+o } is an integral domain <> For h = 1 a,DT , = a k D= J 1 " =>1 ^-s m 6 I^o The lemma holds by induction on Lemmas D <2o^ o 08> H l. J °> I Proofs Case Is < i < js D? „ is the Identity mapping, J- J Case 2s < j < is Assume lemma tea® for - K h s i o j > e Let a b € I i+1 ? a - £ a k x i*l° k=0 20 b - I V^i' N = max(K * M) < aD I+i„j + bD m,j - N m=0 N L (a n D I>j * Vl^ - I (a n + V D £j " (a * b)D lil ? J n=0 n=0 =1 since D. . is a homomorphism by the inductive hypothesise K+M n=0 k+m=n K+M n=0 k+m=n = (a • b)D iii,j For h = the lemma is true by case 1„ By induction on h the lemma is also true for case 2» 2<>5 The Model Domains Starting with a finite field of prime order we define sequences of finite fields and integral domains on adjunctions of the prime field « Since the modeling process will involve the mapping of polynomials in the modeled domains into these finite fields and integral domains , we refer to them as the "model domains e They will be de~ noted by F. and J? as defined in the following development o Definition s F Q = {0 1 9 2, » <> » . t>P ~l} where p Q is a prime <2o5o01> integer in I n „ Definition s Let @ and © be represented by Oo Then <2o5o02> define -1 -1 O § I Q X ^o —- -> Iq where a„b € Xq -> a O b - a O b, -1 More specifically a @ b = a + bs =1 a © b = a ° bo O § F Q X F — > F Q where a„b € I Q »> a O b - (a O b)(mod p Q )o Q Lemmas {F „@ 9 ®} is a field, <2»5o03> Definitions A function M Q mapping I Q into I Q Is <2o5<>0*f> defined by M Q ' s I Q —> I where aM Q §= a (mod p Q ) for a € Iq and I - {b | b = aM o o a € Iq] and p Q is a prime integer « H Ti*£0£ej&s M s X onfo > p <2o5o05> Proofs Given in most texts with a section on number theory, Definitions Define E,, to be a finite field for < j °^° with respect to the operations © „ © o Let p, be a prime polynomial in E. ,[x.]o Define d 4 -r deg.CpJ to be the degree of p„ in the lndeter~ mlnate x°o 5 22 Definition s P. = {a | a € E, ^x.], deg,(a) < d,} for < J <2o5o08> Definition s The operation O representing ® or © is <2o5<>09> defined as follows for < js O s P j X F i " — > Y y a ° D € Fj => a O b = (a O b) (mod p Theorem s {F,,©,®} is a field for < j„ <2o5ol0> Proofs By theorem <2o3o08> o Definition s The fields E, ., in definition <2o5°07> were - Jo unspecified and not necessarily related » We now define E, = F, for < jo Consequently p. € F„ -iCx»] and the operators can be seen to be consistent Theorem " (Pixu^],©,©) is an integral domain* <2o5ol2> Proofs By theorem <2o3o02>» Definition s A function M JD < J, associated with the °° ^ prime polynomial p. € F. ,[xj is defined as follows s M, s F. -.Cx,] — — > I where aM, = a (mod p.) j j 03 J. j p* j J for a € F. jlx,~}. I = {b | b = aM.s a 6 F.. -^Cx,]^ j 23 Theorem? I = F^, < j. P * J <2.5-l4> J Proofs Let a € F, tCx,]- By definition deg.CaCmod pj) < d, ( Therefore deg.CaMJ < d, and aM„ € F„ Q As a ranges over P. JxJ,, all of I is mapped outo Hence I cF„ Let b € P « o Then deg 4 (b) < d 4 o Therefore P* "" 3 j J J bM« = b(mod pj = b £ I and F« el » J J P j J "" P j Theorems M^ s F^Cx^] onTo~> F j<> ° < J <2o5ol5> Proofs By theorem <2<>3<,10> o Lemma, s P 1 c P, ,[xJ t < J <2o5ol?> Proofs By definition of E„ , F, and <2o5oll>o Theorems F £ P. j ° < i < 3 <2o5ol8> Proofs Let h - j - io For h - o P* = Pj i ,5 c F r Assume that for some h > o, p ' r i c j » Then P c j+1 £ F Ex rt 1 b * <2»5ol7> and F^ is embedded in both F,[x J+1 3 and ] since d. > 1„ Hence P, c F 1+l and consequently the theorem holds by Induction on h Q £°rol3^rxs F a ^ s PjCxj^^ < i < j <2o5ol9> Proofs By <2o5<>l8> and <2o5<,17>o Theorem ? F ± c F, c P, ^xJ c I., < i < j <2o5o20> Proofs The first two relations hold by <2o5°l8> and <2<>5ol9>< By definition F Q c I „ implying FqCx-^ - I o^ x l^ = I l° Assume that F. ,[x.] c I. for some j > 1. Then F 1 £ F 1-l^ x l-' i m Plies F, c X, and Fj[x„ +1 ] c I deg,(a) < P < i < j <2o5o21> Proofs Y b € F i,=i» deg-(b) < since the embedding of P. , into F. maps all elements in F, ., into polynomials in x. in F. which are of degree o The proof is completed by <2o5o20> o Theorem s a € P, => aM„ = a <2o5<>22> Proofs a 6 F. => a € F, 1 CxJ by <2o5<>20> and deg,(a) < d.o Therefore aM, = a (mod p.) = a Q Theorem s a b g P. => aOb = aOb 0 Proofs Let h = J - i„ For h = 0, the theorem is true trivially o Assume that for some h > a O b = a © b for a„b £ F l0 a 9 b <= P, c F J+1 by <2o5°2Q> and deg j-fl (a) < deg j+1 (b) < o Therefore a ^ b and i leg holds o (a 6 b) mod p.,, - a b since a © b € b) < Oo By induction on h„ the le Proof? By <2o5°20> and <2o5°22>< The sets J? are defined as follows J <2o5o50> J P 1 tov < j < a ! J®. ,[xj for < a < j ^ J J-1 LX J- Lemma t P .. c J , c I , <2o5o51> Proofs Case Is < j < a by <2o5o20> Case 2.8 < a < . j s Let h •- j - a Q For h b Q„ P, - J, and F. c J. c I. Assume that for some h > F « £ J « £ I*« L ©t a € F 1+ x» Therefore a € F [x. +1 ] c J'^Cx, 1 ] s J^,, by the inductive hypothesis., Hence F^ c J^. J^ c 1^ => J*^ . J^Cx J+1 ] c IjCz^] ^1+1° ^ le ^ emma holds by* induction on ho 26 Lemmas F i->l^ x i-' - J j 9 ° - a < J <2<>5°52> Proofs F., c J? ! by <2<,5o51> and hence Vi Cx j ] £ j j=i Cx j ] = J " Lemma s J J c J* < 1 < J <2o5o53> Proofs Case Is < i < j < as by <2o5°20>„ <2o5ol8>o Case 2s < i < a < js j£ = F ± c F^Ex^] c J^ by <2o5°52>o Case 3s < a < i < js Let h = j - i o For h = J? c J?o Assume that for some h > J i c J. and a € J^o Then a € J * c J^Cx j Proofs First it must be established that (®„$) are valid binary operators on J. X J*» Case Is < j < as J a = F, and a„b € F. => a Q b = a O b by <2o5o23>° Consequently a a a {J „©„©} is a field and hence an integral domain. Case 2s < a < j s Let h = J - a Q For h = 1 £ J aVl e ® 9 J is an integral domain by <2 o 3»02>o Since J? +1 = J ?t x j^i^ f J a+h°®° J is an inte S ral domain by <2o3<>02> and induction on h Q 2o6 Further Conventions on notation In chapters 3 and 4 it will "be convenient to have available a set of conventions about functional notation which will help to reduce the size of expressions and equations involving functions We therefore define the following s Definitions Let A be a set (A) 1 = As A 11 ** 1 « (A) n X A„ ««/ ss I, a a s ififtoSi oS«j o o o j a -» y » a^ c a J o Definition s The component extraction operator K. is <2o6o02> deflned by k s (A) n — > As aK^ s a^ where a = (a^..,,a^,,,,a «)o Definitions The function distribution operator is defined <2o6o03> by q . (A) ny F _^ u ^ p) n whepe P a {f | f g A ~> B}. a € U) n , f € F => (a,f )e s 1&QX Sl=j I p o o o O^L*, -] i Jo Definitions Let f s A — > B 9 a € Ao Then define <2o6 o 0^> a(f) n m a fop n £ 0? a(f)n s a(f} n-l f for n > Oo Conventions It will sometimes be convenient to treat a <2o6o05> binary operator as a unary operator modified 28 by the second in the pair of operands . Let f s A ^ B > C, b a € A 9 b e Bo Define af = (a,b)f „ The following are some examples s (& Q9 & 19 a 2 ) = a € (A) 3 => a(e) K 2 = a £ ffff f o a(er = (a ff a 1 ff B a 2 ff) Definitions i J a = {j£ | < i) <2o6o06> j = {J a | „i < a} (J-domains) I = {I. | < i} = J =1 (modeled domains) F = {F i | < i} (F domains) The sequence (p »P 15 p ? »o«o) of <2o5»0?> will sometimes be referred to as the "prime context sequence" B Definition s U = {Zq ~«>} The symbol « has the following properties i Va € Z Q , °° + a = °°; °° - a = °°; a < °°o 29 CHAPTER III THE RECURSIVE MODELING FUNCTIONS 3ol Introduction At this point we have available a sequence of modeled domains (Iq 9 I,„ <> » ,,1^ „ <> . « ) which are integral do=> mains with respect to the operations (+ °)o Based on a prime integer p Q in I Q we have defined a sequence of F domains (F 00 F 10 „ „ » ,F., « o e ) each of which is a finite field generated by the sequence of prime elements (PqoPto ° <. • p * „ o . where p» is a prime polynomial in Fj..[x.3 for < j The ring operations on the sequence of F domains are repre- 11 j j sented by the operator sequence { (@„©) , (®„©) „ o ° » „ (©, o o o ,? o Based on the F domains sets of sequences of J domains were generated one of which is represented by (P ,F 1M ,, F aol ,F a ,J a+lM ,, J. M ,,) where F Q through F a are finite fields as m the F domains and J- 5 J„ .30°°° r a ,a crt-l" a+2' are integral domains formed by successive adjunctions on F < The corresponding operator sequence is then 11 a-1 a=»l a a a a ( ($„®) o (B S) o o o o „ ( e o G ) ff (e„6) o (9,®)', . - . . ) The object of this chapter is to define sets of functions which map an element from any of the modeled do- mains into any of the F or J domains and which in addi- tion are homomorphismso 30 3=2 The C Modeling Functions Definition s The function C s I i <3«2o01> ■> I, m is defined by the following? I. , = {b b = aC„ „ „ a € I,} Let a € la « Then if i > 0„ we assume a has degree m m x a k x i where a k ^ Ii„i • Define the auxiliary function k=0 C. . § I. > I. . where I. . = {c c = aC. . , a € I,} and the image of a is aC i.J m I (a k C j-l«J-l )x J k=0 m ^1/ ' L"® (a k c i-i^ } k=0 aC i.j-1 for = i = j for < i = J for < J < i for < i < j Then define aC. . = aC. .M. where M Q § I Q — — > F Q ; aM Q = a (mod p Q ) M 3 ! P j-l tx J ] — * P J ! aM J - a(mod V ° * 3 A necessary and sufficient condition that the above definition be valid is that I. Q c I Q and » *i 1 ^ F i»i^ x i-' fo:r ° K «5 ° This condition is shown to h old 31 in Theorem <3o3<>l6>o In addition it is shown that 1^^ , gF, and that C, . is a homomorphism from I, into F,o 3o3 The C Modeling Theorem Proofs The purpose of this section is to prove the fol= lowing two theorems s I ± Q c I Q s 1^ , c F, x [xJ for < J <3<>3ol6> C ioj ° I j "*"* P j < 3o3o25> The above two results are the only ones required for further developments o If the reader is willing to accept their validity he may skip the rest of this section and go to section 3°**-° * < 3 < J J <3o3o01> Ijg mffi&s a aC„ . - a C , J i J , j Proofs a € I* ^> a € I i ^ <2o^ o 03> o Let h = i = J, For h s 0„ the lemma holds trivially* Assume that for some h > 9 aC, , = aC„ |0 a € I* -> a € I 4 , n and i o j Jo J l l'*"- 1 ' 4e« 1+1 U> = 0. a< = I J * «a C ; ) - «< = aC* . k-0 By Induction on h the lemma is true» if <3o3o02> 32 t Proofs C Q is the identity mapping from I Q onto I Q and hence I Q Q = I Q o Let a € I Q o Then aC Q Q = (aC Q )M Q = aM Q £ F Q o Therefore ij Q c F Q , Let b € F Q „ Then b € I Q by <2o5ol8> and bC Q Q = aM Q = a by <2.5<>2^>o Therefore OB F o - I o o° Lemma s I c F^CxJ, I~ c F, for < j <3o3o03> m Proofs For j = 1„ let a 6 I* j a = V a k x.,s a, € I Q o 8 m 3 k=0 aC l„l = I (a k C )x l 6 F [x l ] since a k C 0„0 * F b y k=0 8 8 <3o3°°2>o Hence L , c F £x-. ] since I, ., is mapped out completely as a ranges over I- Let b € I-. • Then bC l 1 € F 0^ x l^ and ^ bC l 1^ M 1 € F l° But bC, , s (bC, 1 )M, 6 I-, -j implying that 1=, , c F, „ Assume 8 _ _ oo for some j > that I* . c F. tLx^J and I. . c F„ k Zk a k x Hl° a k ^ I l° Then aC J+l.J+l - I (a k c 3 ,j )x j + l € F j tx j + 1 ] since k=0 oo aC. , ^ I. . cF. by the inductive hypothesis „ Therefore JoJ J » J "" J z 'i*x,i*i £ f j Cx j+i L (aC j + i„j + i )M J+ i € F j+i ■"* aC j+i.j + i - (aC j + i.J+i )M j € ^l.j+r Therefore oo 1 1+1 . -j c F„ ,„ By induction on j the lemma holds . Lemmas I~^ g Fj <3o3»04> 33 Proofs A combination of <3<,3°02> when j = and <3o3o03> when j > Oo Lemmas c! , = C , < i < j <3o3o05> Proofs Let h = j ° i for some i > 0„ Let a € I,. Then for h = 1 aC, ,.- = aC. . and consequently C. o , = C, . since a can range over all elements in I. o Assume that for some h > 0„ C„ . - C* . „ Then it j i t i c aC i 1+1 ~ aC i 1 ~ ^ aC l 1^ M 1 ~ ^ aC i i^ M 1 by the in(iuctive hypothesis o Since aC. . € F« by <3 o 3o04> and F» c F. by <2»5ol8> we have aC. i € F„ and (aC, . )M. - aC i , by <2o5o2^>o Therefore aC % . +1 = aC i t and C 1 .^ ^ (^ A By induction on h 9 the lemma holds Lemmas b e F« ^> bC„ . = b j j t j <3o3o06> Proofs b £ F „ => b € I, by <2<>5o2Q>o For j = b 6 F Q and bC Q Q = bM (Q = b by <2o5°24> Suppose that for some j > b € F« ^> bC„ . ■ bo Then let a 6 F s .-, s m J Jo J j "*"-•- a s iVj+i 5 ffi < deg j+i (p j+i h a k € F r Then o m A aC j+l.J+l s k I Ca k C j j )x j+l s J Q a k x ^l by the inductlTe hypothesise Therefore aC. +1 „ +1 € F.Cx.^] and o m aC j+l„j+l s (aC j+l P j+l )M j+l s I a k x j+l ■ a since ks=0 m 34 < deg, ,(p. J. The lemma holds by induction on jo oo Lemmas F^ c 1.^ <3»3<»07> Proofs For j = 0, F Q = I~ Q by <3o3»02>o Assume that for oo v k some J > FjCl^j. Let a € F J+1 ; a = £ a k x J+1 ? a k € F r m < 2 e8 j+i (p j+i ) - Then k=0 m aC J + l.j+l= ( I (a k c J.3 )x J*l )M J+l k=0 m " I U k C J,j )x J+l SlnCe m ° < J <3»3o08> m Proofs For J = 1, let a € FqCx^s a = £ a k x l s a k € F k=0 a € Ii by <2o5o20> and m ' k=0 „ k=0 Therefore FqCx-^ Q 1 1 1" Su PP ose that for some J - " f j-i Cx j ] £ x j r Let a € f j [x j+i ]; a = JL a *** +lJ &k € F J k=0 35 Since a € I, +1 by <2„5.20>, aC j+l.J+l - I (a k C J„J )x j+l = X/Vj+l " a fe y <3«3.06>, k=0 , k=0 , Therefore a € ^^t « + x a*" 3 - F i^ x i+i-' £ Ij iJ.i ^-n ° The lemma holds by induction on j _oo Ti'ftTl m ft ° ^ * •* — F jj J 9 J J <3«3o09> Proof? by <3o3o0?> and <3o3oO^>< Lemma s I J j = *J-l^ x ^ f ° < j <3o3ol0> Proofs by <3o3o03> and <3o3o08> o o Lemmgs J i j = F jl° °« i< ^ <3o3oll> Proofs Let a € I 1 for some I > o Then aC, . s aC, „ € F, by <3o3<»05> and <3o3o04> o Therefore 1^ , c F 10 Let b € F^ Then b € I- and bC„ , s= b by <3<,3o06>. D bCj . = bC. . - b 6 I i ^ Therefore F. c I. . o o _ <3o3<>12> Proofs Let h = I = j for some fixed j > 0„ For h = 9 ^1 1 ~ F i=.iL x ^ b y < 3o3olO> Suppose that for some h > 0„ 8 (1 I ± j = F, 1 LxJo Then b 6 1^ => bC ± , 6 F 1=l Cxj D o Let 36 k keO r^ j^l ' aC i+l 1 s )® ^ aC i 1^ ^ F 1-l^ x 1^ by ^ e induc ^ ive hypothesis a € I 1+1 s a = £ a k *i+l s a k € I i° Th « k=0 and by <2o5<>12>o Therefore I i+1 « £ ^i„i^ x i-' ^et b € F 1-l^ x j^ s b = A b k x V ^k € F j-1° But then b € *j - I i'fl by <2o5o20> o Therefore "kl.J - bc j„J " I 'Vj-lJ-l'^ = I„Vj = b * <3 ° 3 ° 01> k=0 k=0 and <3<>3o06>o Therefore b € ^^.n < a nd 'i-v**^ - I i4-1 1 implying that F, ,[xj c I. ., « a The lemma holds by induction on ho oo Lemmas 1^ = F ± , < i < j <3»3ol3> Proofs Let h = J - i for some i > o Case Is h = Os The lemma holds by <3<,3o04>. Case 2s h > Os Let a € I*o Then 9 aC i j = aC i i € F i by < 3°3o05> and <3o3»0*+>. aC i J = ^ aC i J^ M J = ^ aC i i^ M 1 = aC i i by <2 °5»24> Therefore aC l J € F i and I i 1 - F i° Let b € F i° b € I i and bC l„i = b by < 3°3o06>? b € P. by <2o5<>l8>o bC loJ = (bC loJ )M = (bC 1 ^ 1 )M = bM = b by <2o5o24>„ There- fore b € I. * and P. c I? ,. Hence i" . = P. for h > 0<> Lgmmas I 10 = I Q <3o3ol4> 37 Proofs Iq 9 = J by < 3o3o02> o Let a € I Q g I. . aC l = aC 1) " a € I i 9 o by < 3o3o01> o Therefore I Q c I Q . Suppose that for some I > 1^ Q s l Q0 Let a € I aC i*l,0 = L ^ Ca k C i 9 } s I K C i } € X by <2o5o02> k^O k»o " u and the Inductive hypothesise By induction on i Lemmas I iffj - F J9 < j < i <3°3°15> Proof? Case Is = j < i . F 0r i = 0, I <3o3 o 02>o Let a € 1^ Then ac" € I Q by <3o3ol^> and o ° '• i o () o s F o D aC i = (aC i 9 )M € P 0° Hence I i 9 o £ F o° Let b € F o Then b 6 I fl c I by <2o5ol8> and <2 o 4 o 03> bC lo0 = bC 0o£} = b by <3o3o01> and therefore o bC i 9 = * bC i„0^ M s bM " bo Consequently b € I? n P e- I S x i 9 0° Case 2s < j < is Let h . ± j for som j > Oo For h = 0„ I JdJ " F j by < 3«3o09>o Suppose that for soma h > 0. ^ = Fy Let a € PjS a = | ^k, a k i T >i} m < degjipjh a 6 P jC I s I t S I 1+ ~ . < i a m aC i^i.J ■ aC J 9 J - J^Vj-l. j-i)^ - j^ - a by <3o3»01> and <3»3o06>„ Therefore a € F. c F . [ x 1 D J j°l j aC i*l,j ^ P j-lt*j3- Therefore aC^^ = CaC^^ = a M J a by <3o3o01> and <2o5<>24> Hence a € l" « and F c 1°°' 38 oo By Induction on h„ F. c I, . for < j < i. Let b € I, bC i i e F i-l^ x i-' by < 3o3«12> and hence bC ± . = (bC^ .)M € F Therefore I? * c F for < j < io Consequently I. . = F. J- » J J J-oJJ for case 2» t theorem s (I i Q = I Q ) and (1^ , c F, -1 Cxj] 9 < j ) <3o3»l6> Proofs Case Is = j < is by <3„3.1^> Case 2s < J < is by <3o3ol2> Case 3s < i < js I ± . = F, g F. c F, ^x,] by <3o3oll> and <2o5o20> o oo Theorem s I„ . c F, 1„J ~ j <3o3ol7> Proofs Case Is < j < is I* = F. c P. by <3o3ol5> 1 3 J J J Case 2s < i < j s I* = F ± c F. by <3o3ol3>< Lemma s C i i = c i i for ° - 1 - J <3»3ol8> Proofs Let h = j - i for some fixed i > o Case Is h = Os The lemma holds trivially. Case 2s h > Os Let a € L. aC i j = ^ aC i j' M 1 = ^ aC i i^ M i° But aC i i e F i by < 3«3»°9> and F ± c F. since h = j - i > 0„ Hence (aC i ^M = aC i 1 by <2<>5o24>o Therefore aC. . = aC , and C, . = C . „ 39 Lemma s b € I, ^> ^i* = bC j i° < ^ < 1 <3o3»19> Proofs For any i > j > 9 b € I* • bC„ . = CbC, JM, = (bC, ,)M. by <3<,3°l6> and <3 o 3 o 01> 1 o J 1« J j JoJ J J J <3o3°20> Proofs Since I°! . = P. by <3o3o09>„ we need only show that J J „ J for a b € I., aC. . $ bC, > - Ca + b)C. . ■*■ JoJ. JoJ Jo J and aC. , bC„ . - (a • b)-C. . Jo J JoJ J 9 J For j = 0, C Q = M Q j I Q ^L> F Q by <2o5°Q5><> Suppose that for j > 9 Q . § I -£-> F . Let a € K 3.J J J M a : r k s Z, a k x j+l" k=0 a k € I, and b e I J+1° b = m m X j+l S b m € I,. Defi: 3 N t- b ^ £ (a n JS=0 ne N ■ max(K 9 M) Then a • + V )x j+l> a ° b s K+M -HI nsO k+m- (a k n ° V ): n x j+l aC A.- 1+1 D 1*1 <* bC j*i, J+l K = C I (a k C 1 It J +1 .3 ,X J + 1 )M J+1 ® ( ] M )x m ,H J+1 K * = ( Y © (a. C j j )X j+l )M j+l • c £* m-0 (b m m C ) JoJ ^ tl )M 3+1 « J . . k J k-0 ffi^O ^ o -> ° Jo^ N j j n=0 N j = ( Y© (Ca n + b n ^ c i i^ x j+i^ M i+i b y the inductive hypothesis n=0 ■ (a + b, Vi.w • c j*i.j+i G bC j*i.j*i K+M j j . nsrO k*m=n "^^^ ana K*M n=0 k+ms=n = ( £ 4 C £ ® (Ca k - b m ]C j))xj +1 )M j+1 by the inductive hypothesis K+M j - ( I • (t I K • V ]c j,j ,I 5»i ,, w ^^ by the n=0 k+m=n inductive hypothesis = (a ° b)C j4l ^l H Therefore C, +1 « + i § I 1+1 "~ — > F 1+1° By induction on J, the theorem holds « CMaUaax' c t ^ : i A — > p j9 o < i < j <3o3o21> Proof; For < i < J, C. . u C. , by <3 3ol8> Therefore c i c j 8 I i ~" > F i £ F r kl <3»3o22> Proofs For i - C Q Q Is the identity mapping and therefore is a homomorphisnio For i > 0„ let a„b € I* + i s K M a = X a k x i+l° b = X b k x i+l° N ~ max(K M)? a ko b k € 1^- k=0 k=0 o g I Assume that for some i > 9 C. Q s I. > X* q aC l+l 9 ® bC i+l„0 = £® (a k C i,0 } ® I®" (b m C i 0* by <2 °5o02> k^O m=0 ' N » F® Vxo^ b n C i,0 } 1V<2.*.02> ns=0 N »l = £® ([a n + \l c ± q) *y the inductive hypothesis n-0 = (a + b)C* +liQ C =1 t .8 Similarly aC 1+1 Q @ oC i + i - (a ° b)C 1+1 q0 By induction on i« the theorem holds » <3o3°23> Proof? G^ s I jL ^^> F^ 1 Cx^] by <3o3°12>o Let h = i - j for some j > o For h ■ let a € I-} a = Y a^x^s M k=0 a k € I jj]L and b € I y b - £ b m x m , b m € Ij_ r m=0 hz Define N = max(K.M) » J-l i aC J 5 J ® bC J,d = I ® (a k C j=l,j=l )x j S I ® (b m C j-l e j»l )x j k=0 m=0 - I ^ (a n C j-l„j-l ^ Vj-lJ-l' 1 " * <2-5-12> n=0 N . -l " I • (Ca n + b n ] Vl,d~l )x J b y < 3 -3o20> n=0 i = (a + b)C. . J 9 J I 1-1 8 8 Similarly aC* * © bC, . = (a ° b)C. ,„ Assume that for some h = i - j > 0, ^ . s I i -*-> F ^xJo Let a 6 I i+1 ? K M a = I a k x i+r a k e J i and b € x i+i» b = I Vi+r k=0 m=0 b <= I, Define N = max(K„M) m l 8 ^"=1 8 aC i+l j ® bC i+l j rH, » . J-l ? H , » % = J® (a k c i„j ) • L e ( Vi,j> k=0 m=0 = I ® (a n C i j ® b^ J by <3o3ol2>, <2.5*12> n=0 N , -, = ) © ([a + b v.^ c i 4) b y tne inductive hypothesis z_i_ n n 1 j n=0 = (a + b)C° +loJ Similarly ac"^^ © hC^^ - (a • b)C 1+lBj . Therefore the lemma holds by Induction on ho Corollary t C ± ^ s I ± -B-> F y < j < 1 <3o3o2^> Proof s Case Is = j < i C i s J i ~ > V M s X "^ F by < 3°3o22> 9 <2o5o05>o Hence C 1§j - C^ i l ± -*-» F Q by <2.3o20> o Case 2s < j < I < 2o 5ol5>o Hence C^ - c"^ i I J> J > F, by <2»3-20>< <3o3o25> Proofs Case Is < I < j s by <3„3o21>o Case 2s < j < i s by <3o3<>2^>o 3<,4 The D Modeling Functions The F-models as defined by the C modeling fun©° tions are useful in themselves „ but they have the disadvantage of destroying the degree information in polynomials <> Co: sequently we define another function from the modeled doma, into the J domains in such a way that the degree informa- tion of selected Indeterminates Is preserved,, with the addi- tional requirement that the functions be homomorphismso It will be shown that this is accomplished with the D ffi'>i©liag functions o to Definition s Define the function D? 1 . : I. > 1^ , where J i.j.oi € Z QS ^ . = {b I b = aD 1 ,., a € 1^ as m a k x l° a k ^ ^1-1° =0 aC. , for < i < a? < j < a 1 J m k aD i.J I (a k D J-l„ 3 -l^J for 0 « D, . s I. — -> J.o Again,, the proofs presented are not re= quired for the developments in later chapters and consequently can be skipped if the reader will accept the validity of the above theorem o 00 Lemma* I o o o = F o <3o5oOi> Proof 8 Iq o0 c F q since a € I Q => aD^ Q . aC 0p0 € F Q by <3o3o02> o Let b 6 F QO Then bD^ Q = bC Q Q = b by <3o3 o 06>< •'• r o £ 1 0,0° Lemma % a € J* => aD. , = a <3<>5o02> ks Proof Case Is < j < a a € J? c I. by <2o5°53> and aD** , = aC. . = a by <3o3c06>< Case 2 s < a < J Let h = J - a for a fixed a c For h = 0, aD*J . = a by Jo J case lo Suppose that for some h > that b £ J, implies m "~ » Q. OL V" 1 k CL bD. , = b Let a € Jj + i? a = ) a k x Hl ? a k ^ J *° TJlen a € I j+1 and a^ g I . m aD a j+l.j+l Lk d Lj )x m k=0 j+1 JL^*^ k~0 by the induct hypothesis = a By induction on h 9 the lemma holds for this case, Lejamas <3o5o03> T a T a Proofs Let a € J J Then a 6 I. and aD. . J Jo J = a by <3o5oG2>< Hence a € I o o and J , c I o J 9 J J J J Lemmas a € I- => aD^ , = &D* . < j < i J 1»J J J ' <3o5o04> Proofs Case Is < j < i < a a 6 I , c I i => aD^ . - aC t . = aC, . = aD^ , by <3o3°19>< Case 2s < j < a < i Let h = i = a for some fixed a and j a€l i -> a 6 I and deg^a) = by <2o4<,05>o Assume that for some h > = i 46 ,0i T-vCl aD« * - aD, *<> Since a € *j*t» deg i+1 (a) = and aD i*l j = Z® ^ aD l j^ = aD i J = aD J J by the inductive hypothesis o For h=l i = a + 1 and aD^, b aD^ , = aD, , by case 1 Hence the lemma holds for this case by induction on h„ Case 3° < a < j < i Let h b i - j for some fixed j „ For h = ff the lemma holds trivially o Assume that for some h > aD, , =aD. .. a € l°*+i by <2 o 4 o 03> and deg i+ -,(a) b o Therefore aD?., , = aD? . = aD? . by the inductive hypothesis 1+J-oJ l J Jo J Lemmas a € J^ => aD^ = a, < j < i <3o5o05> Proof 8 Case Is < j = i Directly by <3<,5°02> o Case 2 s < j < i a € J? => a € I, by <2o5o53> and aD^" , = aD*? - ■ a by <3o5o04> and <3o5°02> o Lemma £ I^j = Jj <3o5o06> Proofs J ^ c I j , by <3 o 5o03>„ Case Is < j < a D i 1 = c i 1 => *i 1 = X T 1 = F i - J< ? <3o3ol5>o X J J S> J J J J J J J Case 2s < a < j m Let h b j - a.o For h = 1, let a € I a+1 ; a = ) a k x a+l k=0 ^7 a k € x a => aD a+l,a+l = I (a k D £,a )x a+l € J a Cx a+l ] = J a+1 k=0 since a v D°J € J^ "by case lo Assume that for some h > m j j,j £ J r Let a € x j+i ; a = I a k x j+i° Then m aD j + i, j+ i = j^V^.j^i « J j Cx J+ i ] = J j-i sinoe b y the inductive hypothesis a.D . £ J. By induction on h„ the case holds * Hence 1^ , £ J? for < a 9 < j implying I*? , = J. Lsnaa * d£< = dJ 1§ o < i < j <3o5°07> Proofs Let a € I i Case Is 0 Case 2 s < i < a < j Let h = j - a for some fixed i and a For h = 1„ a D« „.! = aD, m - aD. . by case 1» Assume that for some i » a+1 i „ a i i * h > aD. . = aD„ . » Then aD?" . , = aD? , = aD? 1 , and hence the case holds by induction Case 3» < a < i < j Let h = j - i for a fixed a and i„ For h = B the lemma holds trivially,, Assume that for some h > aD i,j = aD i,i* ^ aD l j + l = aD i,j " aD i„i° Lemm a .» ij = j£„ < i < j <3o5o08> 48 Proofs D^j = Dj^ by <3o5 = 07> => l£ , = lj 9i = J* by <3»5o06>. Lgmrn&s 1^ * = J *» < j < i <3<»5o09> Proofs J. g I. . by <3°5<>05>» We must establish that T a T a Case Is < j < i < a In this region D? , = C. . <> Hence I? » = I? . = F. = J, by <3<>3ol5>o Case 2s < j < a < i Let h = I - a for some fixed a» For h = 1„ let a € I +1 , m a = > © (a.) where a v € * a 1 = F i by case 1. Therefore a = V @ (a k ) € F, = J^ by <2o5o23> and <2o5o5^>o Consequently k=0 ■ ^n is c J.o Assume that for some h > 0, I, , c J. a+1 o J — J i o j ~ j m a € I? . -, ,» Then a = Y @ (a,) where a, € I? . c J^ = F. k=0 m •< by the inductive hypothesise Therefore a = > © (a,) € F. = J. k=0 by <2o5o23>, <2o5o5^>o Therefore I? +1 * £ J ? and the case holds by induction on h. Case 3s < a < j < i Let h = i - J. For h = 9 I* , ■ J* by <3o5°04><, Assume Jo J J a * a — - a Then that for some h > 0„ I. , = J.„ Let a £ *i+i i° m n © (a, )i a, 6 I 1 1 £ J.o Therefore k=0 2*9 m a = I® (a k ) 6 J j by <2 °5°5^> and lJ +loJ £ Jj° Hence k=0 I? , S J < f° r & 11 cases. 1 s J J Theorem s I. , c J. <3o5olO> Proofs Case Is < j < is l£ = J^ by <3o5°09><> Case 2s < i < js ij . = jj c jj by <3o5*08> and <2o5»53>o Theorem s d£ s I ± -2-> J^ <3o5oll> Proofs Case Is < j < i < a Since D. . = C. . in this region the theorem is true by 1 9 J 1 9 J <3»3°25>o Case 2s < a < i = j Let h = j - a for some fixed a Let a 9 b € I K , M k 3+1* I a k x j*l' a k €lj and b : I Vj+1 5 b m € V Define k=0 m=0 N = max{K 9 M) and assume that D? , si. -*— > J?c aD J+l * bD J+l k=0 m-0 N N " ( I* (a n D L>4n> ® ( X + (b n D L )x J+l )by <2 c 2 o 30><3o5ol0> n=0 n=0 50 N = 1} (a n D lj®Vj,j )x j + i *<*-*-& n=0 N = T © (Ca n -i- ^ n )D? Jx"^ by the inductive hypothesis n=0 la ♦ b)D^ J 9 J K ~ „ M = L* (a ^J.J )x J+l® I 4 (b m D J 5 J )x J + l *<2.2.30> k=0 m=0 K+M = £ © ( 7 S (a k D^ sJ § b m D j D j ))x j+i b ^ < 3o5ol0> <2„5o5^> n=0 k+m=n K+M , = £ © ( £ ® ((a k • b n )Dj j))xj +1 by the inductive n=0 k+m=n ' hypothesis = y © ([ y (a,, • b )]D? Jx 1 },, by the inductive n=0 k+m=n hypothesis K+M n=0 k+m=n = (a ' b)D j+l,d+l Since D^ a s I a - JL -> j£ by case 1. D a+l 5 a-H % J a+1 JL> J a+l t By induction on h » the theorem holds for this case Case 3°° < j < a < i K M Let a,b € I 1+1 s a = £ V^i * k € ^ and b = £ Vi+1 ! k=0 m=0 51 b 6 I* o Define N = max(K.M) and assume that m 1 D i j s X i ~^~ > J j for some i > a * aD i + i, 3 §bD i + i,j vi*s£,>*l* k=0 K+M = Y § ( £ ® (a k D i c j 0b m D i,j )) by < 3^5 o 09> 9 <2.5o24> n=0 k+m=n K+M = Y © ( y @ ((a k ° "b m )D^ .)) by the inductive hypothesis n=0 k+m=n K+M = Y ® (C Y (a k • b m )]D^ .) again by the inductive n=0 k+m=n hypothesis = (a ' b)D i + i,j By a similar argument aD^ § bD°" = (a + b)D? . Let h = i - cu For h = 1, D^ . s I ± — > J^ since D a „ s I — > J t»y case L By induction on h 9 D i J S I i ™^" > J j f ° r ° - J - a *^ i * Case 4s < a < j < i Let a 9 b £ ^i + i and defined as in case 3* By the same reasoning as in case 3 9 we get 52 aD. , © bD. , - (a + b)D. . aD* , © bD? , = (a • b)D* 1 e J 1»J 1« J This time let h = i - J. For h = 1, D* , s 1^— > J* since D. . s I, > J. by case 2» Case ^ then holds by j J J J induction on h» Case 5? < i < j D* = D* si — > J? c J a by <3,5^0?> and <2 5 = 53> 3^6 The Recursive Modeling Function and its Components Now that we have defined all the domains of interesi and have proven the modeling functions to be homomorphismso i" is convenient to draw all the results together by looking at them in a different manner than that required for the formal definitions and the proofs of the theorems It is shown that D, . can be expressed independent of C„ , and that the C functions are a subclass of the D functions. This rela- tionship allows us to call the D. , function in its new formulation " the recursive modeling function" <> Two auxiliary functions defined as component functions of the D, . func- i » J tion will be Introduced for the purpose of describing a model- ing algorithm when the polynomials being manipulated are in the sum of products form rather than in the recursive form, J. Lemma t C* = D^j, < j < i <3o6o01> =1 ! Proofs Case Is = i ■ js Let a € I Q . aD o 9 o = a ~ aC 9 ft k Case 2s < i = js Let a € I*, a = ) a k x, a k € I^ r j,j - I ( «iP3:io-i ,x j = L (a k c j-i.3-i ) = aC i s k=0 ksO aD J" V (- rJ" Case 3s < j < is Let h = i - j and assume 1=1 ' that for some h > Di , = C. »o Let a € 1jl+i» m a = k l a k X i+l 9 a k € I i° 1-1 A 1=1 i_i ft 1-1 o » aD i + i.j - l • ifl.y - I * ( Vi.j> - aC i*i k^O k^O Therefore the lemma holds by case 1 and 2 and Induction on ho Lemma s C i j = D i i» < i < j <3o6o02> Proofs c[^ = C ioi = D* fl by <3o3o05>< Lemmas D i,j ^ D i,i» ° - J - a <3»6o03> 5^ Proofs Case Is < j < a, < i < j s D^ = C = D^ 1 „ Case 2s 0 5 D. . = D; . . Let a € I = aD i + l ej * < '2-5-23^ k=0 k=0 and the Inductive hypothesis <> The lemma holds by induction on ho Lemmas C„ . = D^ 7M , lo 3 lj J <3o6o05> Proofs Case Is < j < is D^~*M = cl .M = C, ,«, Case 2. 0<1 < i, Vi-) . J>i-\ . ^ = C^ D iIj M j = C i 9 j by < 3°5o0?> 9 <3 o 6 o 03> and <3 o 6 o 02>o Corollary s D^ = D^~*M, <3o6o06> Theorem ; The modeling functions can be rewritten in the <3o6 o 0?> . ,, . following forms 1 m Let a € J° ; for < i let a = Y a^*, a k € J* 1 . k=0 55 aD a i.J r* aD J>3 (1) -1 < a < i < j (2) < 1 = j < a S^-i.j-i^ (3) °**<*- j k=0 m I® (a k D i-l,j } k=0 ,aD i.i W < J < i 4 -1 < a (5) < i < j 1 °° Proofs Case 1 in definition <3 o ^ o 01> (the region < i < a, £ J £ a ) can De replaced by the following? aD a i.J aD L - I s (a k D i.i, J ) k=0 aD„ . = aD„ „ l »i l 8 i »1 = a, = i < J - i < a < j < i < a < i < i < a by the use of lemmas <3*6 o 01> to <3o6 o 06> o After the replacement the D. . functions appear as follows s 1 J 56 ■X = a, = i = j aD^M < j = i < a J j J m aD" 1 3 J < k=0 aD^ i < i < j By the addition of the a = =1 case as defined in <2o^o06> the desired formulation is obtained „ Definition s The modularization modeling function D ° is — M« <3o6ol0> , .. , ^ h > a T h . _-^ T tt *«,. J defined as D M ' % J. - — —> J, for Mo i l ~1 < h < a < i„ < lo Let a € J^o a € J« by <2o5<>51> so we can define aD«° = aD. , o Wj[ 1 t, i Definition s The substitution modeling function D a . . is <3°6oll> , „ 4 ^a T a T a _ _ _. _ v defined as D„. . s J. —— > J„ for -1 < a Ol o J 1 J < j J we can immediately state the following lemmas? Lemma ; dJJ' 01 s J^ — > J^„ =1 < h < a < i D < i „ <3o6ol2> i 57 Lemma s Dg s j£ — > J* -1 < a, < J < i, a < J , <3o6ol3> ii;3 *■»■ D S; a D s, - *5, , D £; a > -i < h < a < j < i. <3o6ol^> * J oJ J The purpose and Use of the above two functions can be seen more clearly if we consider the cases in which a - h+1, m h T* k j = i-lo For a € J** a = ) a k x i " the modu l ar i za 'ti on and sub- k=0 stitution functions reduce tos faM h+1 < h + 1 = i aD M i ,h,h+l / m I (a k D i!li^l )x i 0 h+1 will remain unchanged „ The substitution function states that the coefficients which are in J i=al are simply summedo When the polynomials are represented in a computer by a sum of products form successive application of the above modeling functions may be used to obtain the D. . model By lemma <3°6 1^> the modularization and substitution functions are commutative which implies that an optimum path may exist which maps a modeled polynomial in JT into its model in J. o This path optimization depends in general on the prime context sequence and on the structure of the polynomial being modeled „ 58 3o7 Summary of Domain and Modeling Relations a To summarize the relations among the I,, J. and F sets of integral domains 9 the following schematic can be studied cc . — > »i i 2 3 k — -> i Jq 1 J o h ss J- 1 < i i j; 1 i J l ji J i J l = F j < j < a * 'i 1 4 i 4 J a J = J J-l^j ] -1 < a < j 3 ? ^ *\ >\ >l k Jl x >l *i 4 J 3 The set at coordinate (i a) can be thought of as having an adjunction level of i and a modularity of a, where the adjunction level is the number of indeterminates in the domain and the modularity specifies the indeterminate which is the last one in a finite fieldo In other words „ the integral domain at (i„a) has 1 indeterminates and if a > 0„ the degrees of indeterminates x,,x ?ll .oo x has been reduced by modularization whereas the degrees of the indeterminates x -.pooooX. are still possibly intact after modeling., 59 The above schematic provides a good visualization of neighboring set relations . Define the following symbols? C - Center sets Set about which we are talking - (ego (3^-)) L - Neighbor set to the left of C - (3,0) A - Set above C - (2,1) R - Set to the right of C - (3,2) B - Set below C - (4,1) The set relations then are given by the following % L includes C A C is defined by the adjunction of x, to A, C includes A by an embedding e R is included in C B is an adjunction of Co B includes C by an embedding Let a be an element of Co Then a can be mapped into any of the k neighboring sets by functions which we can give the following names? c to L Reconstruct! on (a € C => a € L) c to A Substitution S i,i-1 c to R Modularization c to B Embedding 60 The mappings from C to A and C to R are the modeling functions and the others are the identity function for the cases we have so far considered All k- mappings are homomorphisms „ The reconstruction mapping which we need is only the identity function and hence information lost during the modeling is not recovered „ In a more complicated situation this information can be reconstructed,, For example,, the Jl to J Q modularization is commonly called residue coding which has been proposed as a number system for arithmetic (7) operations in a computer ' « In this case a number of models (each with a different p~) are formed? the arithmetic pro=> cesses are performed in the model domain and subsequently the result is "reconstructed" by the use of the Chinese re™ mainder theorem . The C to A function is termed a 'substitution" rather than a summation since elements in J i<=1 other than the unit can be substituted for x i to produce the C to A function,, If we substitute 3 € J« -< for x, we are essentially finding the remainder modulo the prime polynomial (x.=3) which is equivalent to the modularization from C to B« But 9 since the practicality of generalizing this to a modularization with a higher degree prime polynomial seems questionable , we refrain from calling the C to A sub- stitution also a modularization,, 61 CHAPTER IV AUGMENTED MODELING kd Introduction The D modeling functions developed in Chapter 3 have the ability of producing models in which the degree infor- mation of selected indeterminates is kept intact However, it will sometimes happen that some of the non-zero coefficients of a modeled polynomial will be in the ideal (Pq,P-, *Pp» <> « <> ,p ) » the kernel of the D, . modeling function. When this polyno- mial is mapped into J., < a < i the corresponding coeffi- cients in the model polynomial will be zero Q If the leading coefficient is in the kernel , the degree of the model polyno- mial is lower than that of the modeled polynomial a circum- stance which can cause an algorithm to produce erroneous re- sults when the internal branching is controlled by the degree information as, for example,, in the division and g.c.d. algorithms, There are a number of possible solutions to this pre- dicament „ The first and most obvious is to use a very large prime for p Q and high degree prime polynomials for the rest of the elements in the prime context sequence. This reduces the probability of producing a non-zero element of the D?" . J- 9 J kernel during a computation to an insignificantly small value . It may then be assumed that the degree of the model will always be the same as the degree of the modeled polynomial and that the production of a zero in the model implies that the corresponding modeled polynomial is also zero. In some instances this approach may be satisfactory , but it is too inefficient „ often dangerous and somewhat inelegant for general use A second solution is to add a "virtual degree" augmentation to the model in order to be able to detect when the leading coefficient of the modeled polynomial is in the kernel of the modelings the condition under which the model will be called "degree reduced" «, This is the approach which is developed in what follows «, It will be shown that the degree information can be treated algebraically in a manner closely related to discrete exponential valuations „ The virtue of this approach is that an algorithm in the augmented model domains is capable of detecting when too much degree information is lost to continue o Often the algorithm can proceed to its natural termination point in which case some information about the result of applying the equivalent algorithm to the modeled polynomial is obtained,, In particular, the "model conclusive g c„do theorem" developed in this chapter shows that the goC d algorithm can be applied in the model domains and zhat the algorithm is capable of determining conclusively that two polynomials are relatively prime if no degree reduced polynomial remainder is encountered 4o2 The Augmented Modeling Domains and Functions In this section we define a set of augmented modeling domains which consist of the J domains to which a "virtual 63 degree" augmentation has been added B Once these domains are available , sets of functions are defined on them which pre- scribe exactly the operations to be performed on the J domain component and the virtual degree component of an element in the augmented domain when the function is applied^ Next we introduce the concept of a ' valuation preserving homomorphic 1 (or VPH) function which is roughly equivalent to stating that a function having this property commutes with the augmented D M modularization modeling function* The usefulness of this concept stems from the fact that composite VPH functions can be generated by argument dependent algorithms composed of VPH functions and defined only on the augmented model do<= mains In other words an algorithm can proceed in the model domain as long as only VPH functions are met and at each point the accumulated composite function has generated polynomials in the model domain which are identical to the model polyno- mials which would be obtained by applying the D M modeling function to the polynomials in the modeled domains which are generated by the equivalent accumulated composite function on the modeled domains. These concepts are not developed to the full generality possible; efforts are concentrated on obtaining a set of definitions and relations which are sufficient to rigorously prove the model conclusive g c do theorem, 6^ Definition s (Extension of the degree function cleg* (ah <4„2.01> a € ^ as defined in <2o4,05>) 6^ s j£ - — > Ui a € J, => a € I^o Define deg^ (a) < i = j < a a 6^ = 7 deg (a) (-1 < a < i = j) or (-1 < a, < j < deg^ (a) < a? < i < j Note that deg* (a) was defined on elements in I, so here we first map the elements from J. into I. and then apply "l the degree function^, Also s for a = -1, a6 = deg, (a) . U U De finition s Define the two functions © and © by the <4.2.02> fo i lowlng! ® i U X U — > U P (a s b) © = max(a b) = a © b u u u © z U X U > U, (a,b) ©=a©b^a + b where U was defined in <2o6e07>. u u Lemma t {U u ©} and {U ©} are commutative monoids with ,2o 3> ag the additive (@) identity and as the u multiplicative (©) identity* t(X u.u. , u v u , u a,b c £ J ± => a @ (b © c) = (a © b) © (a © c)„ 65 Proof i Closure and associativity follow from the definitions and since max((a»b) s c) = max(a» b, c) ■ =°° is the additive identity since Va € J*, a © (~°o) = max(a f =») = a. Similarly u a0O=a+O=a« Distribution follows froms u u u (a b) © (a c) = max(a + b„ a + c = a + max ( b , c ) u u = a O (b © c) , Definition s A function X s R - — > U will be called a <4 2 0k> valuation homomorphism from the ring [R t ® } 0] u into U if va ? b € R 9 (a ® b)X < aX © bX (a Q b)X < aX O bX Under these conditions we denote the function X by X s R — > U, Definition ? The 'Virtual Degree* or V=modeling function is ^' ■> defined by the following? m Y k4o Let a € J^ i a = Y a^xj 66 (1) -1 « a < i « j (2) < i = j < a aVj^ = < ae; 1 , (3) < a < i = j WA < J < a s j < i I ® ^ a k 6 i-l. j } WB " 1 - a < j < i 5) < a s < i < j Define V = {V^ ^ | V^ » Jj 1 — > Us -1 < a, < l, < j} _ _.oc T »l V t» Lemma , ' V, „ § J\ — — > u <4 o 2o05> Proof? For all cases but (4)B (-1 < a < j < i) s aV?" , = a^T 1 ,, — — — — i „ j i 5 k k = i j or «> = u (a + b)v£ . = (a + ^)S^ X k < maxCaS^^bd^) = aV^ . © bV TIT U a • b)vf , = (a • bid? 1 , = ae! 1 , + b6: X v = avj 1 . © bV„ a It J 1»-K 1 9 K 1,K 1 „ J 1 9 J K M Case (4)Bs -1 < a < J < is Let a = T a k x^ ( , b = V b m x m f k=0 m=0 N = max(K,M) aV i,j® bV L= Z^vlii.^5 Z® ( vi-i,j ) k=0 m=0 n=0 67 n=0 = a + b)v! > n=0 k=0 m=0 K+M u xi u -, = I • < I® K 6 i-i ;J °Vl=i.j n=0 k+m=n K+M u u - I • ( I® ([a k * b m ]6 i-lJ }) n=0 k+m=n K±M u _ , > ) 9 (C > (a„ b ra )]6. 1 1 .) L> Li n m 1=1 D j n=0 k-t-m=n - (a « b)V? , Definition? Let b € J" 1 . Define b = bV^" , € U Let a € J* and define a = (a. a) to be the augmenta= tion (or augmented model) of a* a is called the D com= ponent (or D-model) of a and a the V component (or V-model) of a Note that a = aK Q; a = aK x by <2,6 02> Define j£ = {& | a = (a,a)„ a € J^, a € U} , Then ±1 £ Jj X U, It should be observed that the V component is not strictly dependent on the D component. An augmented modeling function will be defined which assigns a value to the V com= ponent of every a € «L » Then sets of functions on L will 68 be defined which will prescribe the exact operations to be performed on :he D and V components when the function is applied-. Hence the two components are related in the sense that the values will always be derived from the same set of modeled polynomials and by the application of a parallel set of functions c The functions defined on J„ will be such — 1 that a and a will always be related by the inequality ct. ° - aS « < £ a ° Because of this property a can be referred to 1 9 1 as the "virtual degree" of a* Definition s B (0 <=«>) 1 = (1,0), ^1 = (-l s 0) <4»2<»07> Definition s The Augmented modeling function is defined by <4o2o08> ~a _»1 . T a _ . ^ _-l • ^ TT , D„ . s J. — — > J.e Let a € J 4 t a € U and —1 J —1 ~J i a - (a,a)o Then aD, « = (aD, , , aV, J : Note that the V ^""1 J 1 D J 1 « J component of a is not used during the modeling- Whenever a modeling is performed a new V component is generated based only on the D component of au Definition s Augmented modularizations D^° a s J^ — > J^ 9 =1 < h < a < 1„ For a ^ £„ ^m = ' a ^M t> a '° Definition ; The augmented addition and multiplication func- tion @^ and ©? represented by O^ are defined 69 as follows? Q£ s (J^) 2 — > j£ c a,b 6 j£ => (a b)o£ = ~a a . u . , aQ^b= (a O b, a O b) Definition s The degree function on J? is defined by <^o2oll> ,a _a . TT x a -a *I k % -i ~"~ > -i° For a = (a„a) € j£«, let Definition s The k 8 th coefficient function; <*K2°12> _a . T a m a = V a,x 1 for 1 > 0» Then the augmentation of a. is k=0 a. — ^ a-. „ a^ / o -i.k ~ A ,_ 9 -<») (=1 = a = i) or (0 < i < a) (aj^xj.c) (=1 = a 9 < i) or (0 < a < i) where c = if a, > Ox c = =« If a, = ~ eo < Definition s Leading coefficient functions <^o2ol3> a o. „ T a ___^ _a T . _ ,a . v £* o J* — — > J„ Let a € J^o k = aK, = a f (0 9 -*») if k = -oo a ^i = ■{ a Definition s Reduction functions <4o2ol4> a / T a*2 . T a T . . r T a P< s (J* ) — > J Let a D b 6 «L 70 a b„ n (c,a=l) if < b < a 9 b = b6 ajg, = (a b)p, = < o -i All other casess (b < 0), (a < b) t«j < b. where c = [((b*f) 0? a) ©? ((-1) 0? (a*?) @? b g° x^)]K, — — 1 ~ 1 "~ —1 —1 — *=- 1 — 1 1 1 V and h - a ° "6o Definitions Residue modulo bs H? s (J?) 2 — > J?o Let <^o2<»15> „ K g. T a 1,9 b € «L < -i v -i' ~i b (a 5 b)R^ - &(p£) , h = a - b + 1 Note that the function is well defined even for h < by <2o6o04> o b is called the divisor of the residue function,, Definitions f ± m {f£ | f^ s (J^) n — > (J^) m -1 < a < i} f, is called a valuation preserving homomorphic (VPH) set of functions on J? and J*} if for a 6 (J?) n and -1 < h < a < is n h D a n h , a 1 a h 1 Define VPH^ c0L = {£ | X ± is a VPH set of functions on -i [? for -1 < h < a ^ i} 71 Lemma ? f,g € VPH^' a => fg Proof s Since D^ f ' a s J^ — > j£ by <3*6*12> the D compos ent equivalence follows immediately <> The V component equality follows since aD» f &.. = a* Lemma s © r ± € VPH^ a <^,2d8> Proofs Let a„b € J^ 6 and O stand for © or © (^^.bD^Q? - ((aD^' a ObD^ a ) 5 (a O b) ) i i i i h h a o u . = £(a O b)D;J t,( \ (a Ob)) by<3 = 6„12> i = (aQ* b)D^ a Lemmas % k g VPH^ t0L , -1 < h < a < i <4o2»19> m h Proofs Let a € J*; a = (a.ah a = Y® ( a k x i^ k=0 Then m a Y© (a,D!j a *- k a k = (a k ,a k ), aD » a = £© ^M^ )x i a ^ D h n cx _ , )D h.a _ f( D h a } } aD ha a _ ( D h 5 a &) a _ (( D h,a . } aii M i ii„k " laL, M i taJ £i s k ~ Ua k D M i ^ 1 Jx i^ c; where c = if a, > and c = »°° if a, = °°° 72 Lemmas ^ € VPH*" a <^.2o20> h c h cu * Proofs Let & € J** k = a = aK, . aD^" K, = a = k= ha ^ k = =>*> -> aDJJ 9 = a = o For k = -«\ a/ h D h ' a = 0l£' a = ^m'^1 - a .-o> = j i " x i ■o i ^ n h_h , a h _,h , a _h , a. a ^h , a a For k > o, a^ D = a* lik 2„ = aE M 2C 1|k = a2 M : *r i l i i i b ~" M i <4,2o21> 11 Proofs The condition < b&^ = b = bD{j° a 6 guarantees that b^* £ and t^ sCt ^ ^ 0: Let a € jj* b ^M ± Case Is a < b ^> a(pj) - a, fl£JJ tCX (pJ) = aD^ a Case 2? a > b. The 'c 9 function in the £^ definition <4,2.-l^> is composed of functions in VPH„ ' and hence is also in VPH h:Cl , The fact that the leading coefficients in b and bD M are not zero guarantees that the coefficient of x a in 1*1-4 1 c will be zero and hence that the virtual degree can be reduced m both domains, vh. a — M i eorems (ifjDJJ' a = D*» a (R*) for b € J £ JJti <4„2<>22> 73 < b6* , = b = bD^ tCX 6? . . , n b m \ Proof? Under the given conditions (p£)DJJ ,a = D{j ,a (P^) Since IL is a composite consisting of only £. functions, the theorem holds,, 4,3 The Model Conclusive g.cd. Theorem We have available now a sufficient number of functions defined on the augmented domain to generate polynomial remainder sequences in the model domains and to develop the model con= elusive goCd, theorem with which we are able to show that two polynomials are relatively prime with respect to a specified indeterminate by performing operations only in the model domains A number of examples are given to demonstrate the power of this theorem and to illustrate some problems which might be en° countered when applying the theorem in practical cases Definition s Let &„b D r € jj, -1 < a < I, < i. Then (a b t r! is called a polynomial remainder triplet (p.rjt.i a ° " a over J. if a6, > b6 > r& and there exist b,cL„ a_ g J. „ Jj = (b 9 0) a = (a ,0) and b 6 = a 6 - such that b a = (b © q) © (a © r) „ It follows that a = b © q. Note that 6 Is used to represent 6„ „ 1 , 1 L = '.EqiLioomL.) is called a polynomial remain* der sequence (pc.rso) if each triplet 7k '£ m <£ n n.r£nrt.2 ) is a p ° rttc over ^i for °- m - n= 2t £ is called a normal p.rcS, if r_ , n 6 = r6 - 1 for 1 < m < n< r is called a complete p rcSc if lJ^-q = ° The above definitions are consistent with those in references (5) and (6)« It is to be noted that the V com= ponents were not used in the definitions and hence a p,r,t, and poToS, could have been defined only on J. • Theorems (Euclidean Algorithm) If r is a complete p r c Sc <4 3 03> aJ " J then r , is an associate of the g c„dc of r nt r,o n^x u x. a Proof? Since J. is an integral domain, r = implies r t divides associates of r for < m < n - 1 ; Hence n-1 m — ~ the goCodo of r n ,r, must be an associate of r , «> 1 n-X Let t- (lmJ,,,...!.) and r_ , n - ( r , r_. . , ) R , a K) 5 ±-l so v ^n } """ ^m+2 " 'Tn^+l'-i' J Then r is a p^r.s. if and only if r Q > r-, and r = r 6 for 1 < m < n. We call r the p ( r,s t generated mm — — by (r ()l E r R 1 ) ( Proofs Under the condition r« > r, and r = r 6, — 1 mm r_ = (r^ ( ,r, )R„ ; r = r n - 1 by definition of the R 4 function ~*c ~"U ~"X "~1 U —i —2 mm antees that r~ = (r^rOfL * By induction on ni;, we see that ^£o°~l°~l^ generates r 75 If r Is generated by (E^r^IL), r Q > r^ by the definition of a generator of a pcT.s. and by the definition of Re If for some k» ^— v-?'— k-1'— k^ is a P° r » to and r v ~ ° or r k & < r k£ , (£ k .!-£ k )Sj = £fc_i' Hence (x k .i«£ k »E k+ i) is not a p»r.to Therefore if r = (rQ,£,,...,rL) is a p<,r„So o and (r A , o o « f r e r .t ) is not, the condition r - r 6 holds —0 —n ~-n+l m m for 1 < m < n. Definition s r = (r Q9 r, „ «, » . ,r ) is called a conclusive p t r t s <^o3o05> if r ^ _ and / or r ^ _ 0c By the p rece cLing lemma and the definitions of R. , a conclusive p.r.s. must also be normal c The above definitions and lemmas hold for polynomials in the modeled and model domains since no restriction was placed on the value of a rj the modularization level. The important problem to be solved is to find a relation between the two which allows us to make a conclusive decision about the modeled polynomials while performing operations only on their models in the model domain. The specific question we =1 wish to answer is whether two polynomials In J. are re= i latively prime with respect to x. s Theorem s Let En*— 1 € J7 p r Q > r-, = r,6 > 0: Then v^o'-l'-i ' generates the p r So r = (r^r,,, „ <, „ r N ) where N > 2. Let s = rD~ 1,a € J?e Then s > L and if s l ~ s l^i 1° ^n 9 — 1°— l^ generates the p<,r s = s - (s A9 St p o o o 9 s ) where n < N Q b Proof s (S 1 ) € VPH^° a if < b = b6 h . = bD^° a 6? , by <^ 2o22> = ~~ ~" ~"" ""1 1 °~" "" 1 , 1 Tl. "~1 o 1 Hence «s 0>Sl )H^= , By induction on m» (s nv s n ,R?) generates s and s., = s =1 for m < n. s„ 6, . = or Q. "• s 6, „ < s t =1 and consequently the p.rrS terminates . -=n i, 1 n=l H * ^ s = r D~ implies s6. , < r 6~ , = r< Hence n < N = — n — n^M, ^ — n i .. i — — n i , i — Theorem s (Model Conclusive g „ c d , Theorem) „ Let r s s be the <4o3o075 p $7iSo of the prev i 0US theorem and let g = gcC.do (rQ 1) r-j)o If s is conclusive 1) with s = 0, s ^ then g6° = 0c 2) with s n > 0„ s n = then g6~ 1 < s -. Proofs Case Is s R = 0„ s n ^ 0s By theorem <^3.06> n < N -1 =1 and hence n = N and r N = o Therefore g6„ . ■ r w 6 i i = ~1 "1 a since r KT 6. , = -«> would imply r = and s = r D M 4 ' = jm 1 1 n n n—n. which contradicts the case hypothesise Case 2s s > P s - 0% r« », is an associate of -1 =1 •=! the goCodo of r Q and r, in J. o Hence r ^^i^i ± ~ s ^i ± and r N = Oo But n < N by theorem <^ o 3o06> implying s n-l = r n4 - r N=l = r N=-l 6 i 9 .i° Stated less succinctly the model conclusive g CocL theorem states that if a model p r s« is conclusive and the last remainder is not 0„ the original modeled polynomials r Q and r., are relatively prime in the indeterminate x.o The second condi= tion states that if the last term in the p„r<>So is 0„ the goCodo of r Q and r^ must have a lower degree in x than the degree of the second last term* The model conclusive g c Codc theorem has much prac- tical significance since it will show that two polynomials are relatively prime when the por.So is conclusive much more rapidly than methods in which a g c d algorithm is applied directly to the modeled polynomials To get a better grasp of the implications of this theorem we can consider the following set of non-trivial examples? ty orations The polynomials will be represented in sequence form as explained in appendix A, 78 Example s Find the GCD of r Q = 1 + x? c , r, = 3 + x-^ °^° •* g = goCodo (r 09 r,)o To illustrate a conclusive PoToSo in which s = 9 we use p« = 5° r Q = 1 1 ,2 r x = 3 1 ,1 1 2 ,1 z 3 = ,0 r~ = implies that g6~ _ < 1 We repeat the calculation using p Q = 7° r Q = 1 1 2 £l = 3 1 »1 14 ,1 r 2 = 3 ,0 This shows conclusively that the two polynomials are rela= 2 tively prime o The problem for p Q = 5 was that 1 + x, factors into (3 + x-,)(2 + x 1 ) modulo 5„ Examples The following polynomials which appeared in reference Wo page 588 illustrate a number of problems which may be encountered and at the same time show how effective the model conclusive g,c 8 d, theorem is, a = (~x 3 )y 6 + (5x 2 )y 5 + (x 3 +x 2 ~7x)y 4 + (~2x 2 ~2x+3)y 3 + (x 3 ~2x 2 °2x*l)y 2 + (x 2 +6x+5)y ♦ (~x 3 ~3x 2 ~3x~D 79 b = iV a Let x 1 = x 8 x ? = y D Then the polynomials are represented by a = * 1 2 3 ..l -3 -3 -1 1 5 6 1 2 1 -2 =2 i 3 3 -2 -2 k -7 1 1 5 5 6 =1 b = 5 6 1 1 2 =4 -4 2 2 9 =6 =6 3 -28 4 4 k 25 5 -6 Let the prime context sequence be (p Q = 5t>Pi Then X-n —J. 9000/0 aS 2.2 * 1 4 2 1 1 2 1 3 3 3 3 4 3 5 6 3 1 1 1 4 3 2 3 2 3 2 3 <2 ft 2 1 1 I 2 1 1 2 2 4 4 4 3 2 4 k h 5 4 £0 I 2,2 *1 = ^2 5 2 ((2 2,3.^0,0,^),6) ((2„l,>2oOp0 9 4) 5) 2 £ 2 s ( £o'£l } £ = ^® ((2,0 2,2,0) 4) SinCe M2.2 3 ^ r ? - 4„ the result is inconclusive „ Now we use the prime context sequence (p Q = 11 P-. = x, + 10 $ o o o s o 80 aD*^ = ((3 B 1 B 9 10 6„5,10) 6) ~2 2 = ^ 1 "7,8 1> 2 P 3 9 5) 9 5) ^0 = - aD 2.2 = 8 10 2 1 5 6 1 06 Xi = 9© (b£ 2j2 ) = 9 8 6 7 5 1 o5 (r^r^P = 8 1 5 6 9 1 >5 £ 2 = ^£o o -l^-°-l^- 2 o r 2 = 3 © r 2 = 10 8 4 1 10 8 10 8 4 1 ,4 ter^ = 9 5 10 8 ,4 £3 = ((£ 1 »£ 2 )£o£ 2 ^- = 3 7 1 o3 (£ 2 9 -3^ = 8 1 5 1 o3 £4 = ( (£ 2 »£3)£o£3)£ £4 = 5 © r^ = 8 7 9 1 9 1 .2 .2 (£ 3 »£i|)£ = 7 6 ,2 £5 = ((£3»£i|.)£»£^)£ s 2 1 .1 (£5"£i|)£ = 7 10 • 1 £* = ((£«ror, l )p 9 r,,)p B 9 ,0 The result is conclusive and indicates that the g c d is inde< pendent of y Q If we try to use the D , model to check if the goCodo is independent of x-. an inconclusive result will always occur no matter what the value of p QO This is caused 3 by the symmetry of the polynomials in the coefficients of x£ which will generate a zero coefficient during summation,, £i ■ p^ 2 1 wl H De degree reduced and consequently cannot be used as a divisor in the reduction function,. 81 If It is still desired to continue in the model do- main r two alternatives remain.- The x-, and x ? indeter- minates can be interchanged by a substitution operation and a prime sequence* (Pq^Pt = x, « p,o„o) where 3^1 can be 1 2 used In the modelings? r Q = &D and r, = bD 2 _, If this still does not destroy the symmetry which causes the cancellation, P-i's of higher degree in x-, can be used: Alternatively,, since it has been established that the g.ccL is independent of y the pair-wise g o Cod* s of the y coefficients can be calculated e This discussion should pro° vide sufficient evidence to Indicate that there exist significant problems in determining the best strategy to be used after an inconclusive p.r.s- is encountered in the model domain o Example t As a final example, we consider a 36 term, 2 vari= J able, total degree 7 pair of polynomials whose coefficients are decimal digits extracted from a table of random numbers, Collins (6) demonstrated that of the h goC.d methods he considered only the reduced p.r«s e algorithm was capable of showing similar pairs relatively prime In a reasonable amount of computation time. The prime context sequence is (p Q s= ?9P-i ■ x-, - !•.••)• 82 a = * 1 £ 1 & i 6 2 Os 8 4 6 7 8 8 6 1 Is 8 8 4 2 8 9 2 2s 8 6 2 3 5 3- 6 7 1 5 4s 5 5 5 5? 9 9 3 6s 7 8 7? 5 O 1 £ Os Is 2s 3s 4s 5s 6s 7s 7 7 8 9 9 4 3 9 7 1 3 o l 6 4 7 5 6 3 7 6 1 8 3 1 9 6 6 6 6 5 6 6 3 2 6 £ = ^2,2 £o = 4 © 4 i £]_ = En © 3 £ 3 (£ 2 ! -3 - ^4 z 5 ^6 ^7 *8 3 6 5 2 5 3 5 5 o 5 2 4 1 1 6 2 3 2 1 4 3 3 o 3 4 5 1 1 Z 1 a 1 £ 2 15 5 2 2 3 4 2 4 6 5 1 3 2 4 3 1 5 6 1 3 4 6 4 5 3 1 1 1 6 3 6 6 1 1 1 2 1 1 1 1 3 1 1 5 l l 7 ,7 r? o7 ,6 ,6 ,5 ,5 ,4 ,4 o3 ,3 2 ,2 ,1 .1 83 The result is conclusive and Tq £ implies that the g.Cod, does not contain a term with x_o It should be noticed that if the two polynomials do have a g.cd, of degree 1 or greater the algorithm in the model domain will always be either inconclusive or have a final remainder of zero in the sequencer s 1 will be a model of the actual goC d* but having available only the developments derived previously in this chapter we are not able to calculate the actual gcCodo from this model „ In a polynomial manipulator the model g.cdo algorithm should be used as a "filter" which is able to catch most polynomial pairs which are relatively prime depending on specific values of elements in the prime con- text sequence o The polynomial pairs which do survive this filtering will very likely have a non-trivial g c d a At this point the filtering can either be repeated with a much more refined prime context sequence (e go p Q is larger) or goCodo algorithms such as discussed in reference (6) can be applied in the modeled domain. kok On the Use of Augmented Modeling in Elimination Let &■. € I and let the equation a, = be represented by e,o The system of equations E = C e k I e k ° a k = ° ° — k — n ~ "*"} then has n equations in m unknowns which are in general not linearly related c 84 The elimination method is the orderly application of the re- duction operator o P» to the equations in the system so as to attempt to reduce the problem to that of finding the zeros of an equation b = where b g L. For the case in which E is a linear system of equations the method reduces to the commonly used Gaussian elimination in which the system E is represented in matrix form. The operations used during elimination are similar to those used in finding the g c d<, of two polynomials and consequently we can expect to be able to generate some con- clusive results by operations performed only in the model domains * As an example we will demonstrate the use of augmented modeling as a method for checking the consistency of the system E when E is linear and overspecif ied in the sense that n > m<» LiQir e = 16a( G-i 6nj 6«/ j e-^. s a-^. = u v a-, c ^-o e Q s x~ + 2x 2 + 3X-L + 4 = e-^ 2x~ + 4x 2 + 5X-L + 6 = e 2 ? x~ + 9x + 6x 1 +2=0 e_s x~ + 3x 3 + 5^ + 1 = e: The above is a system of 4 equations in 3 unknowns in which the initial ordering was made arbitrarily „ Since we will have to be able to Interchange the ordering,, we define the following operator: 85 Definition s S n . s (A) n > (A) n ; S* , induces a permuta- 1 4 J -L J <4 C 4„01> tlon Qn (A) n guch that fQr a = (a ,a,,,o.»a J 6 (A) , aK„ = aS„ .K and aK. = aS^ .K^ where the component function K is defined in <2o6o02>« For convenience we make the following notation conventions e ^ «2 — >6n(6«i i e /-> ^ — n l ' 9 ~ o o 5 °3^ ~ ^SrvoSiu^^vSo The system of equations are represented in the matrix form 1 2 3 4 2 4 5 6 19 6 2 13 5 1 First we generate the D> . model of the system with p Q = 7» The V component of the augmented models appears after each comma; only the V components which will be used in the elimin- ation are shown. 1,1 2,1 3,1 4,0 2 - pD° . 2 « 1 ^ X 5,1 6,0 1,1 3,1 5,1 1.0 The application of the elimination method on the system results in the following set of system states 1 86 1,1 2,1 3,1 fc.O t - (2 o roUc 0,0 0,1 6,1 5<0 degree reduced in x. e - ve^^ ^f < o,0 0,1 3,1 5,0 " " " n * 0.0 1,1 2,1 4,0 , -, 1.1 2,1 3,1 M I - (e e S 3 )• °>° 1 ' 1 2>1 4 *° 2 *1 S 0,1 6,1 5,0 1,1 2,1 3,1 *K0 2 _ ^2 , A t\. 0,0 1,1 2,1 4,0 0,0 0,0 3,1 5,0 o„o o,o 6,1 5„o -~ ( %),i v ~2 5 3 ( -^ c 0,0 o„o 3,1 5,0 i 1,1 2,1 3,1 4*0 e - (2 1 (0))- °»° lsl 2sl ^° e- iSo |2 tS^l£JJ- o„0 5 3,1 5,0 e 0,0 0,0 2,0 4 The last equation in e states that 2 = 0(mod 7)i a contra- 1 11 diction » e. is degree reduced in its equations e-, and e ? * Since e-, should be used as the divisor in the reduction, the method would fail since £ would not be in VPH. By performing an interchange of equations with the operator 3 2 1 S« p ( , e-,= e~ is used as the divisor in the reduction which 3 generates e t As a result the entire sequence of operations performed on e are all in VPH and consequently theorem <4-2 17> applies. Since equation e~ is a contradiction , a corresponding contradiction holds when the equivalent sequence of operations in the modeled domain is evaluated. In other words , the above contradiction in the model domain implies that in the modeled domain the rank of the augmented matrix of the linear system E is greater than the rank of the coefficient matrix. 8? It is of interest to note that the sequence of opera- tions applied in the model system creates an "operation trail" based only on decisions derived from the model system When this same trail of operations is applied to the modeled system it is guaranteed that the divisor of any reduction operation will not be zero and consequently that the reduction will be performed effectively o This in turn implies that as long as the model operations are valuation preserving homomorphic functions,, all sequencing decisions in the elimination algorithm can be made independent of the information in the modeled system,, Though a further study of the applications of augmented modeling has not been made it is expected that its use can be extended to nonlinear systems and to the reduction (2) of systems of differential equations as discussed by Brahns , Also under the VPH proviso the solutions generated in the model domain are models of the solutions in the modeled domain and consequently if a number of model solutions using different primes are generated,, a reconstruction based on the Chinese remainder theorem which generates the solution should be capable of being defined for at least some classes of linear systems o In reference (9) where Moses discusses the prac= ticality of solving systems of polynomial equations by elimina- tion it is brought out that the main problem is the explosive growth of coefficients o Since augmented modelings can be 88 defined so that the coefficients are elements of finite fields, this growth is completely under control. However, the problem that now presents itself involves determining whether or not sufficient information is still retained in the model to draw conclusive results from it* 89 CHAPTER V THE PRIME CONTEXT SEQUENCE 5ol Introduction In previous chapters the existence of prime context sequences (Po»PicPp»° t ">) was tacitly assumed- Since the primes in these sequences determine the properties of the modeling we will briefly look into their characteristics and generation,, The prime integer p Q plays the most Important role since the rest of the polynomials in the sequence are defined on adjunctions for which p Q determines the prime subfieldo 5>2 PO The prime integer p Q determines the number of elements in the prime subfield and therefore can be used to control the amount of storage required for the models a To maximize the information content in relation to the number of bits of storage in a binary machine „ p Q should be of n the form 2 = m where n is the number of bits and m is the smallest odd number which optimizes some other aspect of operations in F Q o These other aspects cannot be specified independent of the particular method and equipment used to perform the operations „ For example „ if the operations are performed by subroutines in a computer with an unalterable structure, not much of an increase in speed can be achieved by choosing different values of m However, if the hardware 90 in the computer is alterable in such a way that it can for example perform ones complement 13 bit arithmetic „ addition and multiplication modulo 8191 can be performed very rapidly whereas a choice of p = 2 - 31 would require much more time to perform the same operations given the identical hardware configuration e This then is an example of hardware being naturally tuned to a particular p Q » If the circuitry in the computer 8 s central pro- cessor is not rigidly specified,, it may be possible to change it dynamically so that it will always be tuned to the particular p Q being used. This in turn implies that addi- tion and multiplication modulo p Q can be performed in about the same amount of time required to perform integer addition and multiplication respectively » That this is possible should be apparent by observing that the end around carry in ones complement arithmetic corresponds to the reduction of a number modulo 2 n - 1. If the prime is 2 n - m 9 the end around carry digit is replaced by the addition of m to the least n bit positions of the number , Showing that this results in a valid modulo p Q operation requires too many details which it is undesirable to introduce at this point . However , we may observe that the circuitry required is minimized if m and consequently p Q has the least arithmetic weight possible where the arithmetic weight is defined as the number of non-zero digits in the canonical 91 (sometimes called the non-adjacent ) binary receding of the number o Primes of the form 2 - m with m as small as possible and of least arithmetic weight are tabulated in appendix B for n = 3 to n - 2*K Though addition and multiplication in modular arithmetic can be relatively simple and fast s the analog of division in integer arithmetic is less trivial since the process requires finding the inverse in F Q of the divisor and multiplying it with the dividend Division and inver- sion do not have the same importance as addition and multi- plication in finite field operations 9 but they are used frequently enough to justify studying the optimization of the relatively complex inversion operation Since a search of the literature indicated that only table look-up has been employed to any extent for inversion, it seems appropriate that the following theorem be presented? Theorem s The following three methods are valid algorithms for the computation of the inverse of a € F Q , lo (Product Method) Define b^ $ ± such that h ± € {0,1}, 3 Q - a 3 ±+1 = & ± © (^ Pi -i n b. and p Q - 2 - ) ^2 Then a x = tt © (^ )° i=0 X 92 2o (Recursive Method) If a = 1, a" = 1. If a ^ 1, define r Q = a and determine n s q. from the equation p = q. • r. -. + r. where n is such that r n = 1. Define = 1, Q i = (~q i+1 ) © Q i+1 for < i < n - lo Then a" 1 = Q QO 3« (Factor Synthesis Method) If a ^ 1, calculate q,, r. , Q. by the iteration p = q Q - a + r Q ; Q Q = r Q + 1 . = q ± • r i _ 1 + r ± i Q i = (Q 1 _ 1 $ (-1) ) © q ±9 i > Find n such that r •, = o Then a" 1 = (-l) n+1 0^9 (~l) n+1 ] (r n ) -1 Proof s Part 1 is based on Fermat's theorem which states that for any non-zero element , a„ in a finite field of P -l order p Q » a = L This implies that the inverse of a P0~2 is a o The proofs for the other two methods depend on Induction applied in a direct manner and so are not included The most interesting of the three methods of in- version seems to be the recursive method since with it there appears to be the ability of optimizing the procedure by choosing an appropriate p Q and it also seems to have the 93 property of requiring fewer operations than the other two in- version methods r particularly if the inversion circuitry is tuned to operations in P Q , Attempts were made to find an analytic result for the maximum and average recursion depth for a particular p Q o This problem appeared to be untractable analytically? so the brute force method in which all non-zero elements in the prime fields generated by selected p Q of the form 2 •= m where n ranged from 3 to 16 were inverted to find the maximum and average recursion depths for these primes «, The results of these calculations again did not indicate an asymptotic value which either of the recursion depths approaches o There does seem to be a correspondence between the maximum recursion depth and a Q „ the number of factors in p Q - 1. The calculations seem to indicate that the more factors p Q = 1 has, the smaller the maximum re- cursion depth can be expected to be 3 In appendix B the maxi- mum and average recursion depths are listed for some of the tabulated p 's. One additional property of the prime subfield which might lead to some interesting results concerns the 2 values of p Q for which xt * 1 is irreducible.. If this is 2 the case , x, ■*■ 1 will be a prime polynomial and the vari- able x-, in the finite field F, will have the property of the complex variable i This points toward the representa- tion of models involving complex numbers in which i is 94 2 represented by x-, o If the polynomial xr + 1 is irreduci' ble„ p - 1 is referred to as a quadratic non-residue (QNR) and if it is reducible to two linear polynomials , a (12) quadratic residue (QR) o This is the reason for pro- viding two primes for each value of n in appendix B<, If a significant portion of a computer's time is to be devoted to polynomial manipulation it is important that the operations modulo p Q be optimized . Of all the operations in various model domains these prime field operations will be performed most frequently «, Consequently it is expected that the speed of the polynomial manipulator will be strongly related to the speed of these prime field operations when finite field based modeling is used, 5o3 p lf 1 > We have already seen examples of p^'s with i > o Specif ically, (PqpP-. = x-, - l,p ? = x 2 - l,„o.) defines an elementary prime context sequence in which each of the fields F. is identically equal to the prime subfield. Although the above is a very degenerate form of prime context sequence,, it is expected to be of great practical value 9 the reasons being that it contains a totally symmetric set of polynomials., If the modeling is performed on a polynomial in several vari- ables and a permutaion of the variables in both the modeled and the model polynomials is performed , the model resulting 95 from the permutation will be identical to that obtained by- applying the modeling function to the permuted modeled polynomial o In this sense the modeling is not affected by the permutation of the variables and of course the new prime polynomial context sequence will be identical to the one used prior to the permutation Since the permutation of variables may possibly lead to procedures for optimizing some polyno- mial operations this symmetry property of a prime context sequence might be worth persuing further for other less degenerate sequences It is also Interesting to note that the above elementary prime context sequence produces re- dundant information which is essentially identical to that (8 ) produced by the function coding used by Mart in <> Finding prime context sequences with polynomials which contain more than a few terms can become quite com- plicated o Fortunately the finite field theory which has been developed provides us with one main theorem which can be used to calculate non= trivial polynomials for the sequences « The theorem is stated In reference (1) theorem 19 „ p Q 135 » what follows is a repnrasal of that theorem using the terminology of the previous chapters „ Definitions Let q ± a p Q ° X 2 ° i where d Q = 1 and <5o3o0i> d„ ■ degjpjo Then a € P- D a £ is said to 96 q i -l be a primitive element of F. if a =1 and there does not exist a t such that t divides (q,-l) and a = 1« Theorem ; Let a be a primitive element of F. and let t - )o - > ° * be an integer such that t divides (q.-l). Let r be another integer such that g c do (r„t) = l c Define g s goCodo (r 9 q,»l)o If 4 divides t we require that t r q. = 1 (mod h) * Under these conditions i. , - a is an t (Qi =-1) irreducible polynomial belonging to the exponent — : — ± — - S The properties of polynomials used in the prime context sequence for practical polynomial manipulation re- quire that multiplication in the finite fields be optimized.. This in turn dictates that the prime polynomials contain as few non-zero terms as possible » The binomials produced by the above theorem are therefore optimum in this respect., There are quite a number of open questions related to finding optimum prime context sequences « As an example we 9 can have one sequence in which p Q = 2 -9»P-. = x-,- 1 and 3 2 3 another in which p Q = 7,p, = x^ - 5 = x^ + 3° The elements of F-. of both sequences contain approximately 9 bits of informations yet the structure of the fields is quite differ- ent The speed of operations performed in F, depends very heavily on the choice made» For example „ if operations in Fq require negligible time compared to polynomial operations 97 in F.. the first sequence should be chosen The relative merits depend in general on the particular methods used to implement the polynomial manipulator . 98 CHAPTER VI CONCLUDING REMARKS The objectives of this investigation were to find a class of redundancies of polynomials in several variables over the integral domain of integers which would be useful in enhancing the power of polynomial manipulators by in- creasing their efficiency or their capability of handling more complex problems than are currently possible „ The finite field based modeling functions of <3„6o07> and <*K2o08> have been shown capable of doing botho The most illustrative example presented is the application of the augmented modeling method to the determination of the greatest common divisor of two polynomials? it is shown that the method can be used to increase the speed of determining whether two polynomials are relatively prime by possibly orders of magnitude o At the same time the method makes feasible the application of the g c d test to polynomials of a size for which it would be impractical to apply even the best avail- able goCodo method without using modeling „ The reason for this large decrease in computational effort is that the coefficients in the model polynomials are finite field ele- ments which implies that the amount of storage and manipula- tive effort required to work with them has a relatively low upper bound o Normally the problem in polynomial manipula- tion is that the size of the coefficients is not bounded and consequently the storage and manipulative effort required can 99 rapidly reach values which make it impractical to perform the manipulation even on a fast computer Though the modeling can be defined so as to produce conclusive results in most practical cases „ it is to be under- stood that there will always be cases in which the results are not conclusive o a fact which must be borne in mind when polynomial manipulations dependent on decisions made from the models are used. Although much of the information in the polynomial can be retained in its model the possibility of decisions being based on lost information does exist and hence the models cannot be manipulated blindly without taking this into account „ The virtual degree augmentation to the modeling is one example of a technique which may be used to prevent this type of decision error Q Inconclusive results may still be encountered,, but with the augmentation they are detectable, making it possible to take some remedial actions such as reordering the variables „ using a different modeling or even changing the prime context sequence o If none of these actions produce a conclusive result one can always return to the modeled domain and apply the algorithm without using modeling provided of course that this is a practical possibility o It is postulated that the same techniques as used in the g c d calculation can also be successfully utilized to reduce the computational complexity in elimination and in 100 the reduction of systems of differential equations; results in this area are expected to be particularly worthwhile in regard to making algebraic manipulation efficient enough to be of more general use than is now the case Some additional areas in which modeling might prove useful are polynomial factoring 9 error checking,, operation optimization by model determined orderings and printed polynomial output monitoring c There is also the possibility that the modeling can be applied to functions other than polynomials in a manner analogous to (8 ) the coding used by Martin D 101 APPENDIX A On the following page a polynomial in I~ with 64 terms of one decimal digit each is mapped Into all its possible model domains under the prime context sequence 2 P = 7 P-, = x^lfi p 2 = Xp-1 p~ = x~-lo The information is compressed in such a way that the polynomial x-l + (2+3x^)x 2 + ( 4x^+5x^2+6x2 )x 3 + (7x 1 +(8+9x^)x 2 )x^ would be represented by 00 1 1 ll 2 3 10! 4 1 5 2 6 20 1 0?0 l| 8 9 An X in a position indicates that the coefficient which should be present is greater than 9o 102 * indicates the start of a polynomial in J ? * 00 X * 1 ° 1 2, 3 00 X X X X * 1 2 3 001 X 8 X X 1 X X X X 2 X X X X 3| 7 X X X * 12 3 00 1 2 3 6 2 6 6 3 2 1 3 6 7 9 19 5 8 10 1 2 3 30 l 2 3 3 2 3 3 9 7 6 2 5 5 2 4561 20 j 6 3 9 2 1 1H5 2 10 7 1 31 3 ^ 5 0383 6267 8 2 2 2 7 7 5 OOi 1 00 1 2 3 10 1 2 3 20 1 2 3 30 1 2 3 12 3 ool 0366 * 1 2 3 00 1 1 1 1 1 2 k 1 2 5 4 3 3 1 5 1 2 3 6026 6321 3602 1251 3 2 3 3 2062 5 5 2 4561 6322 1001 03^5 3 13 6260 1022 2005 P = 7 00 00 1 2 3 10 1 2 3 20 1 2 3 30 1 2 3 1 6 2 » 1 00 2 1 1 5 3 2 5 4 3 1 l 1 1 6 1 4 3 1 6 3 6 5 1 2 5 3 6 1 5 5 2 1 1 k l 1 6 5 2 3 2 2 5 P x = x^+6 00 20 1 00| 6 2 1 10 1 6 k 2 30 4 1 1 k OOI 6 2 P 2 = x 2 +6 3 P. x^+6 103 APPENDIX B Minimum Weight Primes p - 1 i s a QNR P - 1 i s a QR n m w Q MR AVR n m w a MR AVR 3 1 2 4 2 lol7 3 3 2 3 4 5 3 4 4 lo90 4 3 3 6 3 io50 5 1 2 8 5 2o30 5 3 3 6 4 2o43 6 5 3 4 7 4o29 6 3 3 12 6 2o63 7 1 2 12 7 3o34 7 15 3 10 9 4oll 8 5 3 8 12 5o79 8 15 3 20 8 3o62 9 9 3 4 14 7o97 9 3 3 6 11 5o6l 10 5 3 4 16 8<>52 10 3 3 24 13 5o60 11 9 3 4 17 9o24 11 31 3 36 12 5o30 12 5 3 8 15 7o09 12 3 3 24 15 6o60 13 1 2 48 15 6ol7 13 31 3 48 16 6 78 14 65 3 8 22 10o57 14 3 3 72 16 6o73 15 49 4 32 20 9*95 15 19 4 12 21 9o39 16 17 3 16 24 11»71 16 15 3 120 19 7o88 17 1 2 32 17 31 3 144 18 5 3 8 18 11 4 18 19 1 2 64 19 31 3 48 20 5 3 32 20 3 3 96 21 9 3 4 21 31 3 16 22 17 3 4 22 3 3 72 23 1 2 32 23 15 3 10 24 5 3 8 24 3 3 48 i 2 n ~ m w = arithmetic weight of p Q o Q ss number of factors in P ~l MR = maximum recursion depth in recursive inversion method AVR - average recursion depth 104 REFERENCES Cl] Albert,, A<> A , " Fundamental Concepts of Higher Algebra", UniVo of Chicago Press, Chicago Illo, 1956 Q C2] Brans, Co Ho, A computer program for non-numerical testing and reduction of algebraic differential equations 9 J ACM 14, 1 (Jan 196?) p Q 45-62 [3] Brown, W D S 09 Hyde, Jo Po and Tague, B D Ao , The ALPAK system for non-numerical algebra on a digital computer—I Is Rational functions of several variables and truncated power series with rational function coefficients , Bell Sys Techo Jo 43 (March 1964) , p Q 785-804 o [4] Collins o Go, PM, a system for polynomial manipulation,, Commo ACM 9o 8 (Augo 1966 2 „ 5?8-589o [5] Collins, Go Eo B Polynomial remainder sequences and determinants, Amer Matho Mon Q 73 7 ( Aug „ -Sept Q ) Po 708-712o [6] Collins, G Eo, Subresultants and reduced polynomial remainder sequences, Jo ACM 14, 1 (Jan c 1967) p 128-142 o [7] Garner, H« L , The residue number system, IRE Trans on Electro Comp op EC-8, (June 1959) Po 140-147 « [8] Martin, W Q Ao, Hash-coding functions of a complex variable. Artificial Intelligence Project Memo 70, MIT, Cambridge, Masso 1964 [9] Moses, Jo, Solution of systems of polynomial equations by elimination, Comm ACM 9, 8 (Aug, 1966) p 634-637o [10] Paley, H Q and Weichsel Po M op "A First Course in Modern Algebra", Preliminary Edition, Holt„ Rinchart and Winston* Inc e , New York, 1963o C.11] Van Der Waerden, B c L , "Modern Algebra", Volo I, Ungar Publishing Co OB New York, 1953 o [12] Vinogradov, Io M Q , "Elements of Number Theory", Dover Publications, Inc O0 19 5^ ° 105 VITA Stephen John Nuspl was born in Stanisic, Yugoslavia on February 25 » 19^0 » He received a B AoSc<> degree in Electrical Engineering from the Assumption University of Wlnsor„ Windsor „ Ontario in June, 1963° Since then he has attended the University of Illinois and has held a research assistantship In the Department of Computer Science o He is a member of Phi Kappa Phi 9 the Institute of Electrical and Electronics Engineers and the Association for Computing Machinery o ■" JUN 2 01969