A Pommaret Bases Approach to the Degree of a Polynomial Ideal

Co-author(s): Bentolhoda Binaei and Amir Hashemi

Reference: Applicable Algebra in Engineering, Communication and Computing, 29 (2018) 283-301

Description: In the first half, we provide novel proofs for some classical results about Hilbert series and Hilbert polynomials using Pommaret bases. These lead also to explicit formulae for Hilbert series, polynomial and the degree of an ideal in quasi-stable position. In the second half, these results are applied to complexity theory. We derive first a dimension dependent Bézout bound for the degree improving previous results by Masser and Wüstholz. Then we improve a dimension dependent bound for the representation problem due to Mayr and Ritscher which in turn leads to improved bounds for the membership problem and for the degrees of Gröbner bases.

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