Upper bounds for Betti numbers from constraints on the Hilbert function

TL;DR

This paper presents an algorithm to compute sharp upper bounds for the total Betti numbers of saturated ideals based on Hilbert function constraints, implemented in Macaulay2.

Contribution

It introduces a novel algorithm and software implementation for determining maximal Betti numbers under specific Hilbert function constraints.

Findings

Algorithm effectively computes sharp upper bounds.

Implementation available in Macaulay2 package MaxBettiNumbers.

Provides tools for constructing ideals with maximal Betti numbers.

Abstract

We describe an algorithm for finding sharp upper bounds for the total Betti numbers of a saturated ideal given certain constraints on its Hilbert function. This algorithm is implemented in the Macaulay2 package, MaxBettiNumbers, along with variations that produce ideals with maximal total Betti numbers.