Assessment 2 knuth book

To avoid coefficient growth altogether we again use

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To avoid coefficient growth altogether, we again use the idea of Algorithm M. Instead of solving the given problem directly, we solve one or more related problems in Zp[x~, • • • , x~_~], and then reconstruct the desired result. Let Fi' and F ' 2 be viewed as polynomials in g[x~, • • • X~-l], where 9 denotes the polynomial domain Zp[xv]. Although F~' and F( are primitive as elements of Zp[xl, • .. , x~], they need not be primitive as elements of 9[x~, • • • , x~_~]. Let c~, c2, c, F1, F2 , G, H1, H2 , f~ , f~ , g, h~ , h~ , O, G, lP~ , F~ , H~ , and/t~ be defined as in Algorithm M. Now, however, all of the lower ease symbols denote polynomials in a = Z~[x~], while the upper ease symbols denote polynomials in ~l[xl, • • • , X~_l]. For any fixed b ~ Zp, let g~ denote the field of polynomials in ~l modulo the ir- reducible polynomial (x~ -- b). Since for polynomials f ~ ~, the quantity f(x~) mod (xv -- b) is equal tof(b) ~ Z~, we see that ~ is precisely Z~. Algorithm P is essentially identical to Algorithm M, except that v is replaced by v - 1, Z is replaced by 9, the prime p ~ Z is replaced by the irreducible polynomial (x~ - b) ~ g, and Z, is replaced by :¢~ = Zp. In this situation, (31) is replaced by F(~) F1 mod (x~ b~), 1 = (49) while the polynomials 0 (i), H~), and Zp[xl , ... , x~-l]. Furthermore, (34 ~(1) and when e = d, we have for i = 1, /~2 mod( x, - bi), /~(i) 2 satisfy (32) and (33) in gb~[Xl, "" • , x~-i] = ) becomes 0 mod (x~ - b~), (50) ~(i) __- =- • " , n, in place of (35). q = II (xv -- bl), i=l 0 mod (xv - bi), /71 mod (x~ - bd, (51) /I~ mod (x~ - bl), Also, (36) becomes (52)
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494 w.S. BROWN while G*, Hi*, and H2* [see (37)] are the unique polynomials (in ~[Xa , "" , xv-,]) with eoeffieients (in 9) of degree (in Xv) less than n, such that G* ~ G (i) rood (xv - bi), H,* ------ //~) rood (x, - bi), (53) H2* ~ " (i) 1/12 mod (x, - b~), fori= 1,...,n. Now as soon as e = d, weseefrom (51)and (53) that (38 holds. When we also achieve n > , = max (0v (0), o~ (/it,), 0~ (/72) ), (54) where 0v denotes the degree in x,, it follows that (40) holds. To obtain the final results, we then use (41) and (42) as in Algorithm ~[. Although the preceding discussion is sufficient in principle to define Algorithm P, the interested reader may find it instructive to compare the following detailed description with the earlier presentation (Section 4.3) of Algorithm M. (1) If v 1, then F, and F' = ' 2 are elements of 9 invoke Algorithm U to comput G' = gcd(F,', F(), and return. Otherwise use Algorithm U to compute 0 = eont(Fl' ) c2 = cont(F2'), c = gcd(o, c2). (2) Set F~ = Fl'/C,, F~ = F2'/c2. (3) Set fl = lc(F,), f~ = lc(F2), 0 = gcd(f,, f2). (4) Set n = 0, e = min (~(F,), O(F~)). (5) Set ~, = 0~(0) + O~(F,), ~2 = 0~(0) + O~(F2), ~ = max(p,, P2). It follows that h = 0~ (f,) = 0~ (G) + 0v (/4,), ~2 = 0~ (f~) = 0~ (G) + 0~. (/t2), and v > ,. (6) Let b be a new element of Z, such that (x~ - b) ~ fir2. If Zp is exhausted, then p is too small and the algorithm fails. (7) Set 0 = 0 mod(x~ -- b), IP~ = OF, mod(x, - b), F2 = 0F2 mod(x~ -- b). (8) Invoke Algorithm P reeursively to compute G = g" ged (F~, F2), lq, = IPl/G, and tq2 = F2/G, all in 9b[xl, "" , X~_l] = Zp[xi, ... , xv ,]. These relations imply thatle(G) = g, andl)(G) > d. (9) If it(G) = 0, set G = 1, Hi = Fx, He = F2, and skip to Step (15). If i} (G) > e, go back to Step (6). If i} (G) < e, set n = 0, e = ~ (G).
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