Hilbert Functions of Chopped Ideals

TL;DR

This paper studies chopped ideals, which are derived from homogeneous ideals by fixing generator degrees, focusing on their ability to define the same points and their computational complexity related to Hilbert functions, with implications for tensor decomposition.

Contribution

It introduces conjectures on Hilbert functions and regularity of chopped ideals, proving many cases and linking these to practical tensor decomposition applications.

Findings

Conjectured Hilbert function values for chopped ideals.

Proved conjectures in numerous cases.

Relevance to symmetric tensor decomposition.

Abstract

A chopped ideal is obtained from a homogeneous ideal by considering only the generators of a fixed degree. We investigate cases in which the chopped ideal defines the same finite set of points as the original one-dimensional ideal. The complexity of computing these points from the chopped ideal is governed by the Hilbert function and regularity. We conjecture values for these invariants and prove them in many cases. We show that our conjecture is of practical relevance for symmetric tensor decomposition.