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 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 . 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⩾0antn 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.