Symposium on Symbolic and Algebraic Manipulation 

Abstract: In this paper we have tried to demonstrate how sparse techniques can be used to increase the effectiveness of the modular algorithms of Brown and Collins. These techniques can be used for an extremely wide class of problems and can applied to a number of different algorithms including Hensel's lemma. We believe this work has finally laid to rest the bad zero problem.

Abstract: We present a new criterion that may be applied in an algorithm for constructing Grobner-bases of polynomial ideals. The application of the criterion may drastically reduce the number of reductions of polynomials in the course of the algorithm. Incidentally, the new criterion allows to derive a realistic upper bound for the degrees of the polynomials in the Grobner-bases computed by the algorithm in the case of polynomials in two variables.

Abstract: This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to the problem of finding the roots in GF(p) of certain other polynomials over GF(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.

Abstract: Algebraic simplification is examined first from the point of view of a user who needs to comprehend a large expression, and second from the point of view of a designer who wants to construct a useful and efficient system. First we describe various techniques akin to substitution. These techniques can be used to decrease the size of an expression and make it more intelligible to a user. Then we delineate the spectrum of approaches to the design of automatic simplification capabilities in an algebraic manipulation system. Systems are divided into five types. Each type provides different facilities for the manipulation and simplification of expressions. Finally we discuss some of the theoretical results related to algebraic simplification. We describe several positive results about the existence of powerful simplification algorithms and the number-theoretic conjectures on which they rely. Results about the nonexistence of algorithms for certain classes of expressions are included.

Abstract: This paper examines the computation of polynomial greatest common divisors by various generalizations of Euclid's algorithm. The phenomenon of coefficient growth is described, and the history of successful efforts first to control it and then to eliminate it is related. The recently developed modular algorithm is presented in careful detail, with special attention to the case of multivariate polynomials. The computing times for the subresultant PRS algorithm, which is essentially the best of its kind, and for the modular algorithm are analyzed, and it is shown that the modular algorithm is markedly superior. In fact, the modular algorithm can obtain a GCD in less time than is required to verify it by classical division.