The Veronese Construction for Formal Power Series and Graded Algebras

Details The Veronese Construction for Formal Power Series and Graded Algebras The Veronese Construction for Formal Power Series and Graded Algebras

2007-12-17

arxiv.org/abs/0712.2645v1

Mathematics

Combinatorics

Scientific paper

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.

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