On the Hilbert series of ideals generated by generic forms

TL;DR

This paper proves Fröberg's conjecture for ideals generated by many generic forms of the same degree and explores related conjectures about powers of forms, advancing understanding of Hilbert series in polynomial rings.

Contribution

It establishes the conjecture for large sets of generic forms and proposes a new conjecture relating powers of forms to generic forms, verifying it in multiple cases.

Findings

Proves Fröberg's conjecture for large numbers of generic forms of equal degree.

Proposes and verifies a conjecture relating powers of forms to generic forms.

Provides new cases where the conjecture holds, expanding its validity.

Abstract

There is a longstanding conjecture by Fr\"oberg about the Hilbert series of the ring $R/I$, where $R$ is a polynomial ring, and $I$ an ideal generated by generic forms. We prove this conjecture true in the case when $I$ is generated by a large number of forms, all of the same degree. We also conjecture that an ideal generated by $m$'th powers of forms of degree $d$ gives the same Hilbert series as an ideal generated by generic forms of degree $md$. We verify this in several cases. This also gives a proof of the first conjecture in some new cases.