Proof: The upper bound is a combination of Propositions 85 and The first half of this inequality is quite useful and will be used again. The following proposition estimates the size of the factors of a polyno- mial, and is quite useful for polynomial GCD and factoring algorithms. The Pk X are relatively easy to compute from the P X. Recall from Section 9.

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Proof: The upper bound is a combination of Propositions 85 and The first half of this inequality is quite useful and will be used again. The following proposition estimates the size of the factors of a polyno- mial, and is quite useful for polynomial GCD and factoring algorithms. The Pk X are relatively easy to compute from the P X.

Recall from Section 9. By repeating this process we can get P4 X , Ps X , etc. The lower bound is only slightly more tricky. By Taking square roots and replacing P by Pk in this inequality gives D The lower bound of Proposition 88 is not particularly sharp, but the upper bound can be made quite sharp, by increasing the size of k. To illustrate this technique consider the following polynomial suggested by [43] which has zeroes Only the first two zeroes are outside the unit circle and their product is 7.

Figure Then Proof: By Recently, this observation was made substantially more precise by using the weighted L2 norm. Size of a. Proposition 90 Let f and 9 be polynomials in one variable of degree m and n respectively. The following proposition [16] is an analogue of Proposition 89, but uses the [,12 norm. Let Q X be a divisor of P X , also with rational integer coefficients. If this is not the case, we can monicize the polynomial and apply Proposition One of the most useful expressions involving the difference of zeroes of P X is the discriminant of P, which is, essentially, the square of the product of differences of pairs of zeroes of P X.

This quantity is quite useful in the study of algebraic extensions. We write P X as Its derivative is Evaluating at ai we have where the circumflex indicates a term in the product that is omitted. By the definition of the

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## Effective Polynomial Computation - Richard Zippel

Popular passages Page — Frontiers in Applied Mathematics. The eight papers in the book fall into three groups. The Xippel of Melbourne Library. Skickas inom vardagar. Computer Algebra and Parallelism Login to add to list.

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## Effective Polynomial Computation

About this book Introduction Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth.