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17-04-2020 | Original Paper

# A partial characterization of Hilbert quasi-polynomials in the non-standard case

Authors: Massimo Caboara, Carla Mascia

## Abstract

In this paper, we present some work towards a complete characterization of Hilbert quasi-polynomials of graded polynomial rings. In this setting, a Hilbert quasi-polynomial splits in a polynomial F and a lower degree quasi-polynomial G. We completely describe the periodic structure of G. Moreover, we give an explicit formula for the $$(n-1)$$th and $$(n-2)$$th coefficient of F, where n denotes the degree of F. Finally, we provide an algorithm to compute the Hilbert quasi-polynomial of any graded polynomial ring.
Literature
1.
Bavula, V.V.: Identification of the Hilbert function and Poincarè series, and the dimension of modules over filtered rings, Russian Academy of Sciences. Izv. Math. 44(2), 225 (1995)
2.
Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge University Press, Cambridge (1998) CrossRef
3.
Bruns, W., Ichim, B.: On the Coefficients of Hilbert Quasipolynomials. Proc. Am. Math. Soc. 135(5), 1305–1308 (2005)
4.
5.
Caboara, M., Mascia, C.: On the Hilbert quasi-polynomials for non-standard graded rings. ACM Commun. Comput. Algebra 49, 101–104 (2015)
6.
Dalzotto, G., Sbarra, E.: Computations in weighted polynomial rings. Analele Stiintifice ale Universitatii Ovidius Constanta 14(2), 31–44 (2006)
7.
Dalzotto, G., Sbarra, E.: On non-standard graded algebras. Toyama Math. J. 31, 33–57 (2008)
8.
Dichi, H., Sangaré, D.: Hilbert functions, Hilbert-Samuel quasi-polynomials with respect to $$f$$-good filtrations, multiplicities. J. Pure Appl. Algebr. 138(3), 205–213 (1999)
9.
Herzog, J., Puthenpurakal, T.J., Verma, J.K.: Hilbert polynomials and powers of ideals. In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 145, No. 3, pp. 623–642. Cambridge University Press (2008)
10.
Hoang, N.D., Trung, N.V.: Hilbert polynomials of non-standard bigraded algebras. Mathematische Zeitschrift 245(2), 309–334 (2003)
11.
Lee, D.: On the power-series expansion of a rational function. Acta Arithmetica 62, 229–255 (1992)
12.
Decker, W., Greuel, G. M., Pfister, G., Schönemann, H.: S ingular 3-1-6—A computer algebra system for polynomial computations. http://​www.​singular.​uni-kl.​de (2014). Accessed Apr 2020
13.
Vasconcelos, W.: Computational methods in commutative algebra and algebraic geometry, volume 2, Springer Science & Business Media (2004)
Title
A partial characterization of Hilbert quasi-polynomials in the non-standard case
Authors
Massimo Caboara
Carla Mascia
Publication date
17-04-2020
Publisher
Springer Berlin Heidelberg
Published in
Applicable Algebra in Engineering, Communication and Computing / Issue 1/2022
Print ISSN: 0938-1279
Electronic ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-020-00423-1

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