The Veronese Construction for Formal Power Series and Graded Algebras

Abstract

Let $(a_n{)}_{n\ge 0}$ be a sequence of complex numbers 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 transformation of the coefficient series of $h(t)$ to that of $h^{<r>}(t)$ where ${\sum }_{n\ge 0}{a}_{nr}{t}^n=\frac{h^{<r>}(t)}{(1-t{)}^d}$. 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 ${\sum }_{n\ge 0}{a}_n{t}^n$ 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.

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