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

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
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### References:

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