The Veronese construction for formal power series and graded algebras

Let (a(n))(n >= 0) be a sequence of complex numbers such that its generating series satisfies Sigma(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 Sigma(n >= 0) a(nr)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 Sigma(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 Lambda is the Stanley-Reisner ring of a simplicial complex Delta then this relates to the rth edgewise subdivision of Delta-a subdivision operation relevant in computational geometry and graphics-which in turn allows some corollaries on the behavior of the respective f-vectors. (c) 2009 Elsevier Inc. All rights reserved.