Let (a(n))(n >= 0) be a sequence of complex numbers such that its generating series satisfies Sigma(n >= 0) a(n)t(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 Sigma(n >= 0) a(nr)t(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 Sigma(n >= 0) a(n)t(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 Lambda is the Stanley-Reisner ring of a simplicial complex Delta then this relates to the rth 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. (c) 2009 Elsevier Inc. All rights reserved.

Brenti, F., Welker, V. (2009). The Veronese construction for formal power series and graded algebras. ADVANCES IN APPLIED MATHEMATICS, 42(4), 545-556 [10.1016/j.aam.2009.01.001].

The Veronese construction for formal power series and graded algebras

BRENTI, FRANCESCO;
2009-01-01

Abstract

Let (a(n))(n >= 0) be a sequence of complex numbers such that its generating series satisfies Sigma(n >= 0) a(n)t(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 Sigma(n >= 0) a(nr)t(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 Sigma(n >= 0) a(n)t(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 Lambda is the Stanley-Reisner ring of a simplicial complex Delta then this relates to the rth 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. (c) 2009 Elsevier Inc. All rights reserved.
2009
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/03 - GEOMETRIA
English
Edgewise subdivision; h-Vector; Hilbert series; Real-rootedness; Unimodality; Veronese algebra
Brenti, F., Welker, V. (2009). The Veronese construction for formal power series and graded algebras. ADVANCES IN APPLIED MATHEMATICS, 42(4), 545-556 [10.1016/j.aam.2009.01.001].
Brenti, F; Welker, V
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/27052
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