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Assessment 2 knuth book

Theorem 1 let f and f be given nonzero primitive

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THEOREM 1. Let F~ and F: be given nonzero primitive polynomials in Z[x~ , . • • , x,] , let G = gcd(F1, Fo), and let d~ = O~(G), where O~ denotes the degree in X~. Also, let ( i) Sj (F1, F2) :lenote the jth subresultant of F~ and F~ viewed as univariate polynomials in xi with polynomial coe~cients, and let ~ be the (integer) content of ~(~) (F~, F~) *Jdi viewed as a multivariate polynomial with integer coe~cients. (Here :l~ is the degree of G in X~, and not, as in Section 3.5, the degree of the ith polynomial in a PRS. ) Finally, let o" = l~I ¢i. (44) i=1 Then every unlucky prime divides ¢. PROOF. For fixed p from Step (6) of Algorithm M, let P~ = Fi rood p and F~ = F~ rood p. Also, let G = gcd (F1, F2) in Z,[x~, ... , x,,], and let e~ = di(G). If p is unluckv, then there is some i with di < e~, and therefore S (~) (P~, d i F~) ~ 0 rood p. Now let Cl and c~ denote the leading (polynomial) coefficients of F~ and F: respectively, relative to x~ If p ~ oct, then S (~) (F1, F~) S (~) (F~ F:) , d i ------ di , rood p. Otherwise, suppose p divides the first k leading coefficients of F~ for some c(i)(F~ F:) ~ ~,~i)(Fi F~) rood p. In either k > 0. Then it can be shown that ~,d~ , C: ~d~ case t.Jdi~*(i)(Fi, F~) --= 0 mod p, p [ ~(i)~di/F~ 1, F2), p i o'i, and finally p [ a, as was to be shown. THEOREM 2. Let u be the number of unlucky primes p > a, where a >_ 2 is a given integer. Let c = max [~[, (45) where ~ ranges over the coefficients of F1 and F~ . Let v m = ½ max (Oi(F~) + O~(F:)). (46) i~1 Let v t = max h, (47) i~1 where t~ is the maximum number of terms in any polynomial coe~cient of F1 or F2 viewed as univariate polynomials in x~. Finally, let ~7 = my log, (2mc2t 2). (48) Then u < ~. PROOF. Let P be the product of the unlucky primes p > o~. Since P I o by Theorem m /,~ 2.2 \mv 1, wehavea ~ < P < a. Nowby (21),o-~< (2mc2t 2) ,soo-_< ~2mc~) .Itfollows that u < log, a _< ~, as was to be shown. THEOREM 3. The probability that p is unlucky is at most v/p.
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Euclid's Algorithm and Computation of Polynomial GCD's 493 PROOF. For all sufficiently small problems, we have o-~ < p for i = 1, ... , v, and therefore p cannot be unlucky. However, in the general case, we may assume that the quantities ~ mod p are independent random variables in Zp. Hence the proba- bility that p [ ~isp-~ and the probability that p [cris 1 - (1 - p-~)~ < vp -1 where this last inequality follows by induction on v. This completes the proof. 4.5 ALGORITHSI P. Let Fi' and F2' be given nonzero polynomials in Zp[xi, .. • , x~], where p is a fixed, sufficiently large prime. [If p is too small, the elements of Zp may be exhausted by Step (6) of the algorithm.] Algorithm P computes G' = ged (F~', F/), H~' = F(/G', and H( = F(/G'. Since Zp is a field, Fi' and F2' are automatically primitive, and G' is monie. In the univariate ease, Algorithm P simply invokes Algorithm U, which is pre- sented in Section 4.7. In the multivariate ease, we could single out a main variable, and then apply Algorithm C. If so, we would still be faced with the problem of coefficient growth, since the polynomial coefficients would grow in degree even though their integer coefficients (in Zp) could not grow in size.
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