Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

TL;DR

This paper establishes new bounds and asymptotic behaviors for Gorenstein Hilbert functions, proving unimodality in certain cases and generalizing previous conjectures and results.

Contribution

It provides explicit bounds on Gorenstein h-vectors and asymptotic formulas for their entries, extending prior work and confirming conjectures.

Findings

Lower bounds for entries of Gorenstein h-vectors in terms of previous degree

Unimodality of Gorenstein h-vectors in fixed codimension and degree range

Asymptotic formulas for minimal entries in Gorenstein h-vectors

Abstract

We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanley that all Gorenstein $h$-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the $i$-th entry of a Gorenstein $h$-vector may assume, in terms of codimension, $r$, and socle degree, $e$. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case $e=4$ and $i=2$, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree $i= \lfloor \frac{e}{2} \rfloor $.