The Veronese construction for formal power series and graded algebras

Abstract

Let ( a n ) n ⩾ 0 be a sequence of complex numbers such that its generating series satisfies ∑ n ⩾ 0 a n t n = h ( t ) ( 1 − t ) d for some polynomial h ( t ) . For any r ⩾ 1 we study the transformation of the coefficient series of h ( t ) to that of h 〈 r 〉 ( t ) where ∑ n ⩾ 0 a n r t n = 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 ∑ n ⩾ 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 monotonically increasing with r. If A is the Stanley–Reisner ring of a simplicial complex Δ then this relates to the rth edgewise subdivision of Δ—a subdivision operation relevant in computational geometry and graphics—which in turn allows some corollaries on the behavior of the respective f-vectors.