Two counterexamples for power ideals of arrangements

TL;DR

This paper provides counterexamples to a conjecture about the generation of power ideals in hyperplane arrangements and shows that their Hilbert series are not solely determined by matroid structure for certain parameters.

Contribution

It disproves Holtz and Ron's conjecture and demonstrates that the Hilbert series of power ideals can depend on more than the matroid for specific values of k.

Findings

Counterexamples disprove the conjecture for k = -2.

Hilbert series of C_{A,k} are not matroid-invariant for k ≤ -6.

Power ideals' structure varies beyond matroid data in certain cases.

Abstract

We disprove Holtz and Ron's conjecture that the power ideal C_{A,-2} of a hyperplane arrangement A (also called the "internal zonotopal space") is generated by A-monomials. We also show that, in contrast with the case k \geq -2, the Hilbert series of C_{A,k} is not determined by the matroid of A for k \leq -6.