Assessment 2 knuth book

In comparing the primitive prs algorithm and the

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In comparing the primitive PRS algorithm and the subresultant PRS algorithm, we must compare the cost of computing the primitive part at each step with the advantage of having possibly smaller coefficients. In most cases, the primitive-part calculations represent a substantial fraction of the total effort in the primitive PRS algorithm, and there is little if any compensating advantage. However, there may conceivably be some cases in which the subresultant PRS diverges so far from the primitive PRS that the primitive PRS algorithm is actually faster. 4. The Modular Algorithm 4.1 INTRODUCTION. Let F be a nonzero polynomial with integer coefficients. Letf denote the leading coefficient of F, and let p be a prime which does not divide f. Then if F is irreducible over the integers modulo p, it is also irreducible over the integers. This fact has long been exploited by those interested in polynomial fac- toring. Similarly, let F1 and F2 be nonzero polynomials with leading coefficients fl and f:, and let p be a prime which does not divide fl or f2 • Then if F~ and F2 are relatively prime over the integers modulo p, they are relatively prime over the integers. Even if F1 and F2 are not relatively prime, the computation of their GCD over the integers modulo p may provide useful information concerning their GCD over the integers. We shall develop this idea into a general algorithm (Section 4.3) for computing the GCD of univariate or multivariate polynomials over the integers. 4.2 BASICCONCEPTS. Let 9 be a unique factorization domain in which GCD's can somehow be computed, and let 9[xl, • • • , xv] denote the domain of polynomials in xl, • • • , xv with coefficients in 9. When v > 1, we shall not view the elements of this domain as polynomials in xl with coefficients in 9[x2, • • • , xv]. Instead, we shall generalize the concepts of Section 2 to apply directly to multivariate polynomials. Let the exponent vector of a term be the vector of its exponents, and define the lexicographical ordering of exponent vectors d = (dl, - - • , dr) and e = (el, • • • , ev) as follows. If d/ = el for i = 1, • • • , v, then d = e. Otherwise, let j be the smallest integer such that dj # e~. If dj < ej, then d < e, while if dj > e~, then d > e. As- suming that the terms of a polynomial F are arranged in lexicographically decreasing Journal of the Association for Computing Machinery, Vol. 18, No. 4, October 1971
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Euclid's Algorithm and Computation of Polynomial GCD's 489 order of their exponent vectors, the first term is called the leading term. The coef- ficient of the leading term is called the leading coe~cient, and is denoted by lc (F). The exponent vector of the leading term is called the degree vector, and is denoted by (F). Note that a(FG) = ~ (F) + ~ (G). A polynomial is called primitive if its nonzero coefficients are relatively prime; in particular, all polynomials over a field are primitive. The content and primitive part are defined exactly as in Section 2.4, and it is again true that the GCD of two poly-
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