How to compute the Hilbert depth of a graded ideal

TL;DR

This paper introduces two efficient algorithms for calculating the Hilbert depth of graded ideals, especially squarefree lex ideals, and uses them to provide counterexamples to existing conjectures.

Contribution

It presents novel algorithms for Hilbert depth computation and applies them to challenge previous conjectures in the field.

Findings

Algorithms effectively compute Hilbert depth for graded ideals.

Counterexamples to Shen's conjectures are constructed.

Algorithms perform well on squarefree lex ideals.

Abstract

We give two algorithms for computing the Hilbert depth of a \emph{graded ideal} in the polynomial ring. These algorithms work efficiently for (squarefree) lex ideals. As a consequence, we construct counterexamples to some conjectures made by Shen in \cite{B:Sh2}.