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Practical computations with Gröbner bases

Daniel Lichtblau
Organization: Wolfram Research, Inc.
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This is an old manuscript that I abandoned around 2001. There is very little theory, and that small amount (on polynomial factorization) is incomplete. Timings are from machinery of that vintage.

Since their invention by Buchberger in 1965, Gröbner bases have become a pervasive tool in computational mathematics. In this paper we show several sorts of Gröbner basis computations in Mathematica. We discuss computations of approximate Gröbner bases, with applications to solving systems and implicitization. We also show how to use the built-in GroebnerBasis function to compute polynomial greatest common divisors over various fields, series inverses, syzygy modules, and more. Some of this is by now classical; the emphasis is on effective use of off-the-shelf technology, although possibly some of the methods are new. We will point out areas where existing functionality needs enhancement, and note some directions for future work.

*Mathematics > Algebra > Field and Ring Theory
*Mathematics > Algebra > Polynomials

Groebner bases, polynomial algebra, gcds, factorization, enumerative geometry, primitive elements
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computing_with_Groebner_basies.nb (330.9 KB) - Mathematica Notebook