The Veronese construction for formal power series and graded algebras. (English) Zbl 1230.05299

Brenti, Francesco; Welker, Volkmar

Adv. Appl. Math. 42, No. 4, 545-556 (2009).

Summary: Let \((a_n)_{n\geq 0}\) be a sequence of complex numbers such that its generating series satisfies \(\sum_{n\geq 0} a_n t^n= {h(t)\over (1-t)^d}\) for some polynomial \(h(t)\). For any \(r\geq 1\) we study the transformation of the coefficient series of \(h(t)\) to that of \(h^{(r)}(t)\) where \(\sum_{n\geq 0} a_{nr}t^n= {h^{(r)}(t)\over (1- t)^d}\). 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 \(\sum_{n\geq 0} a_nt^n\) 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 \(A\) then this relates to the \(r\)th 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.

MSC:

05E40	Combinatorial aspects of commutative algebra
13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13F55	Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes

Keywords:

Veronese algebra; Hilbert series; \(h\)-vector; real-rootedness; unimodality; edgewise subdivision

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