Algorithms for strongly stable ideals

TL;DR

This paper introduces three algorithms to generate all strongly stable ideals with specific properties, aiding research in algebraic geometry and combinatorics, and analyzes their computational complexity.

Contribution

It presents novel algorithms for constructing strongly stable ideals with prescribed Hilbert polynomials and functions, advancing computational methods in algebraic geometry.

Findings

Algorithms successfully generate all targeted strongly stable ideals.

Complexity estimates for the algorithms are established.

Applications include studying Hilbert schemes and Betti numbers.

Abstract

Strongly stable monomial ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, in this note we present three algorithms to produce all strongly stable ideals with certain prescribed properties: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. We also establish results for estimating the complexity of our algorithms.