Enumerative g-theorems for the Veronese construction for formal power series and graded algebras

Abstract

Let (an)n⩾0 be a sequence of integers such that its generating series satisfies ∑n⩾0antn=h(t)(1−t)d for some polynomial h(t). For any r⩾1 we study the coefficient sequence of the numerator polynomial h0(a〈r〉)+⋯+hλ′(a〈r〉)tλ′ of the rth Veronese series a〈r〉(t)=∑n⩾0anrtn. Under mild hypothesis we show that the vector of successive differences of this sequence up to its ⌊d2⌋th entry is the f-vector of a simplicial complex for large r. In particular, the sequence (h0(a〈r〉),…,hλ′(a〈r〉)) satisfies the consequences of the unimodality part of the g-conjecture. We give applications of the main result to Hilbert series of Veronese algebras of standard graded algebras and the f-vectors of edgewise subdivisions of simplicial complexes.