The Veronese Construction for Formal Power Series and Graded Algebras

Let (an)n≥0 be a sequence of complex numbers such that its generating series satisfies ∑ n≥0 ant 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 anrt 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 ant n is the Hilbert series of a standard graded k-algebra A. If in addition A is CohenMacaulay then the coefficients of h〈r〉(t) are monotonely increasing with r. If A is the Stanley-Reisner ring of a simplicial complex ∆ then this relates to the rth edgewise subdivision of ∆ – a subdivision operation relevant in computational geometry and graphics – which in turn allows some corollaries on the behavior of the respective f -vectors.