Algorithms for Computer Algebra pp 429-471 | Cite as

# Gröbner Bases for Polynomial Ideals

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## Abstract

We have already seen that, among the various algebraic objects we have encountered, polynomials play a central role in symbolic computation. Indeed, many of the (higher-level) algorithms discussed in Chapter 9 (and later in Chapters 11 and 12) depend heavily on com putation with multivariate polynomials. Hence, considerable effort has been devoted to improving the efficiency of algorithms for arithmetic, GCD's and factorization of polynomials. It also happens, though, that a fairly wide variety of problems involving polynomials (among them, simplification and the solution of equations) may be formulated in terms of *polynomial ideals*. This should come as no surprise, since we have already used particular types of ideal bases (i.e. those derived as kernels of homomorphisms) to obtain algorithms based on interpolation and Hensel's lemma. Still, satisfactory *algorithmic* solutions for many such problems did not exist until the fairly recent development of a special type of ideal basis, namely the Gröbner basis.

## Keywords

Computer Algebra Infinite Sequence Polynomial Ideal Quotient Ring Ideal Basis## Preview

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