Normalization of monomial ideals and Hilbert functions

Villarreal, Rafael

doi:10.1090/S0002-9939-08-09182-X

Normalization of monomial ideals and Hilbert functions
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by Rafael H. Villarreal PDF

Proc. Amer. Math. Soc. 136 (2008), 1933-1943 Request permission

Abstract:

We study the normalization of a monomial ideal, and show how to compute its Hilbert function (using Ehrhart polynomials) if the ideal is zero dimensional. A positive lower bound for the second coefficient of the Hilbert polynomial is shown.

References

Martin Aigner, Combinatorial theory, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Reprint of the 1979 original. MR 1434477, DOI 10.1007/978-3-642-59101-3
Adrián Alcántar, Rees algebras of square-free Veronese ideals and their $a$-invariants, Discrete Math. 302 (2005), no. 1-3, 7–21. MR 2179232, DOI 10.1016/j.disc.2004.07.025
Arne Brøndsted, An introduction to convex polytopes, Graduate Texts in Mathematics, vol. 90, Springer-Verlag, New York-Berlin, 1983. MR 683612, DOI 10.1007/978-1-4612-1148-8
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
W. Bruns and R. Koch, Normaliz, computing normalizations of affine semigroups, 2003. Available from: ftp.mathematik.Uni-Osnabrueck.DE/pub/osm/kommalg/software.
Winfried Bruns, Wolmer V. Vasconcelos, and Rafael H. Villarreal, Degree bounds in monomial subrings, Illinois J. Math. 41 (1997), no. 3, 341–353. MR 1458177
A. Capani, G. Niesi and L. Robbiano, CoCoA: A system for doing computations in commutative algebra. 1998. Available via anonymous ftp from lancelot.dima.unige.it.
D. Delfino, A. Taylor, W. V. Vasconcelos, N. Weininger, and R. H. Villarreal, Monomial ideals and the computation of multiplicities, Commutative ring theory and applications (Fez, 2001) Lecture Notes in Pure and Appl. Math., vol. 231, Dekker, New York, 2003, pp. 87–106. MR 2029820
M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. MR 304376, DOI 10.2307/1970791
Bernhard Korte and Jens Vygen, Combinatorial optimization, Algorithms and Combinatorics, vol. 21, Springer-Verlag, Berlin, 2000. Theory and algorithms. MR 1764207, DOI 10.1007/978-3-662-21708-5
Thomas Marley, The coefficients of the Hilbert polynomial and the reduction number of an ideal, J. London Math. Soc. (2) 40 (1989), no. 1, 1–8. MR 1028910, DOI 10.1112/jlms/s2-40.1.1
Claudia Polini, Bernd Ulrich, and Wolmer V. Vasconcelos, Normalization of ideals and Briançon-Skoda numbers, Math. Res. Lett. 12 (2005), no. 5-6, 827–842. MR 2189243, DOI 10.4310/MRL.2005.v12.n6.a5
Alexander Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR 874114
Wolmer Vasconcelos, Integral closure, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. Rees algebras, multiplicities, algorithms. MR 2153889
Rafael H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001. MR 1800904