Enumerative $g$-theorems for the Veronese construction for formal power series and graded algebras

TL;DR

This paper investigates the properties of coefficient sequences of Veronese series derived from formal power series and graded algebras, showing they satisfy certain combinatorial and unimodality properties for large r.

Contribution

It establishes that the difference vectors of these sequences form f-vectors of simplicial complexes, linking algebraic series to combinatorial topology.

Findings

Difference vectors are f-vectors of simplicial complexes for large r

Sequences satisfy unimodality conditions related to the g-conjecture

Applications to Hilbert series and subdivisions of simplicial complexes

Abstract

Let $(a_n)_{n \geq 0}$ be a sequence of integers such that its generating series satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \geq 1$ we study the coefficient sequence of the numerator polynomial $h_0(a^{<r >}) +...+ h_{\lambda'}(a^{<r >}) t^{\lambda'}$ of the $r$\textsuperscript{th} Veronese series $a^{<r >}(t) = \sum_{n \geq 0} a_{nr} t^n$. Under mild hypothesis we show that the vector of successive differences of this sequence up to the $\lfloor \frac{d}{2} \rfloor$\textsuperscript{th} entry is the $f$-vector of a simplicial complex for large $r$. In particular, the sequence satisfies the consequences of the unimodality part of the $g$-conjecture. We give applications of the main result to Hilbert series of Veronese algebras of standard graded algebras and the $f$-vectors of edgewise subdivisions of simplicial complexes.