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

Let $(a_n{)}_{n\ge 0}$ be a sequence of integers such that its generating series satisfies ${\sum }_{n\ge 0}{a}_n{t}^n=\frac{h(t)}{(1-t{)}^d}$ for some polynomial $h(t)$. For any $r\ge 1$ we study the coefficient sequence of the numerator polynomial $h_0(a^{<r>})+\mathrm{.}\mathrm{.}\mathrm{.}+h_{{\lambda }^{\mathrm{\prime }}}(a^{<r>})t^{{\lambda }^{\mathrm{\prime }}}$ 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 $\lfloor \frac{d}{2} \rfloor$\text{textsuperscript}thentryisthe$f$-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.