The Veronese Construction for Formal Power Series and Graded Algebras

Let $(a_n)_{n \geq 0}$ be a sequence of complex numbers 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 transformation of the coefficient series of $h(t)$ to that of $h^{< r >}(t)$ where $\sum_{n \geq 0} a_{nr} t^n = \frac{h^{< r >}(t)}{(1-t)^d}$. We give a precise description of this transformation and show that under some natural mild hypotheses the roots of $h^{< r >}(t)$ converge when $r$ goes to infinity. In particular, this holds if $\sum_{n \geq 0} a_n t^n$ is the Hilbert series of a standard graded $k$-algebra $A$. If in addition $A$ is Cohen-Macaulay then the coefficients of $h^{< r >}(t)$ are monotonely increasing with $r$. If $A$ is the Stanley-Reisner ring of a simplicial complex $\Delta$ then this relates to the $r$th edgewise subdivision of $\Delta$ which in turn allows some corollaries on the behavior of the respective $f$-vectors.