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.


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
Full Text: DOI arXiv


[1] Backelin, J., On the rates of growth of the homologies of Veronese subrings, (), 79-100
[2] Beck, M.; Stapledon, A., On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series, Math. Z., in press · Zbl 1230.05017
[3] Bruns, W.; Herzog, J., Cohen – macaulay rings, (1993), Cambridge University Press Cambridge · Zbl 0788.13005
[4] Brun, M.; Römer, T., Subdivisions of toric complexes, J. algebraic combin., 21, 423-448, (2005) · Zbl 1080.14058
[5] Brenti, F.; Welker, V., f-vectors of barycentric subdivisions, Math. Z., 259, 849-865, (2008) · Zbl 1158.52013
[6] Comtet, L., Advanced combinatorics, (1974), Reidel Dordrecht
[7] Edelsbrunner, H.; Grayson, D.R., Edgewise subdivision of a simplex, Discrete comput. geom., 24, 707-719, (2000) · Zbl 0968.51016
[8] Eisenbud, D.; Reeves, A.; Totaro, B., Initial ideals, Veronese subrings, and rates of algebras, Adv. math., 109, 168-187, (1994) · Zbl 0839.13013
[9] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (2004), Springer New York
[10] Conca, A.; Herzog, J.; Trung, Ngô Viêt; Valla, G., Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces, Amer. J. math., 119, 859-901, (1997) · Zbl 0920.13003
[11] Freudenthal, H., Simplizialzerlegung von beschränkter flachheit, Ann. of math., 43, 580-582, (1942) · Zbl 0060.40701
[12] Diaconis, P.; Fulman, J., Carries, shuffling and an amazing matrix, Amer. Math. Monthly, in press · Zbl 1229.60011
[13] Grayson, D.R., Exterior power operations on higher K-theory, K-theory, 3, 247-260, (1989) · Zbl 0701.18007
[14] Hibi, T., Flawless O-sequences and Hilbert functions of cohen – macaulay integral domains, J. pure appl. algebra, 60, 245-251, (1989) · Zbl 0728.13005
[15] Holte, J., Carries, combinatorics and an amazing matrix, Amer. math. monthly, 104, 138-149, (1997) · Zbl 0889.15021
[16] Reiner, V.; Welker, V., On the charney – davis and neggers – stanley conjectures, J. combin. theory ser. A, 109, 247-280, (2005) · Zbl 1065.06002
[17] Stanley, R.P., Hilbert functions of graded algebras, Adv. math., 28, 57-83, (1978) · Zbl 0384.13012
[18] Sturmfels, B., Gröbner bases and convex polytopes, Univ. lecture ser., (1996), Amer. Math. Soc. Providence, RI · Zbl 0856.13020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.