Betti numbers of piecewiselex ideals

TL;DR

This paper generalizes a result relating Betti numbers and Hilbert functions for certain ideals in polynomial rings to fields of any characteristic, providing a method to compare algebraic invariants.

Contribution

It extends a previous result to polynomial rings over any characteristic, introducing a way to compare Betti numbers of specific ideal combinations.

Findings

The constructed lex ideal matches the Hilbert function of the original ideal.

Betti numbers of the combined ideal are at least as large as those of the original.

The result holds in polynomial rings over fields of any characteristic.

Abstract

We extend a result of Caviglia and Sbarra to a polynomial ring with base field of any characteristic. Given a homogeneous ideal containing both a piecewise lex ideal and an ideal generated by powers of the variables, we find a lex ideal with the following property: the ideal in the polynomial ring generated by the piecewise lex ideal, the ideal of powers, and the lex ideal has the same Hilbert function and Betti numbers at least as large as those of the original ideal.