On Hilbert functions of general intersections of ideals

TL;DR

This paper investigates upper bounds on the Hilbert function of the intersection of two ideals after a general coordinate change, extending Green's hyperplane section theorem to a broader context.

Contribution

It generalizes Green's hyperplane section theorem to provide new bounds on the Hilbert function of ideal intersections under general coordinate transformations.

Findings

Established new upper bounds for Hilbert functions of ideal intersections.

Extended Green's hyperplane section theorem to more general settings.

Provided theoretical framework for analyzing ideal intersections under coordinate changes.

Abstract

Let I and J be homogeneous ideals in a standard graded polynomial ring. We study upper bounds of the Hilbert function of the intersection of I and g(J), where g is a general change of coordinates. Our main result gives a generalization of Green's hyperplane section theorem.