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An effective characterization of complete monomial ideals in two variables. (English) Zbl 1428.13012
The author characterizes complete (integrally closed) monomial ideals in two variables over a field in two ways. First he obtains a geometric characterization using the Newton polyhedron of the given ideal $$I$$. The geometric characterization is used to obtain a criterion based on linear inequalities of the exponents of the monomial generators of $$I$$, thus answering a question of P. Gimenez, A. Simis, W. Vasconcelos and R. Villarreal [P. Gimenez et al., J. Commut. Algebra 8, No. 2, 207–226 (2016; Zbl 1353.13006)]. Here is a quote from this paper: “We hope that this criterion would give some hints for the study of normal monomial ideals in several variables.”
##### MSC:
 13B22 Integral closure of commutative rings and ideals 13B25 Polynomials over commutative rings
##### Keywords:
complete monomial ideal; lattice point; Newton polyhedron
Normaliz
Full Text:
##### References:
 [1] Binh, H. N. and Trung, N. V., The Bhattacharya function of complete monomial ideals in two variables, Comm. Algebra43 (2015) 2875-2886. · Zbl 1354.13013 [2] W. Bruns, T. Römer, R. Sieg and C. Söger, Normaliz, a computer program for computations in affine monoids, vector configurations, lattice polytopes, and rational cones, http://www.home.uni-osnabrueck.de/wbruns/normaliz/. [3] V. Crispin Quiñonez, Integral closure and related operations on monomial ideals, Ph.D. thesis, Stockholm University (2006). · Zbl 1116.13006 [4] Quiñonez, V. Crispin, Integral closure and other operations on monomial ideals, J. Commut. Algebra2(3) (2010) 359-386. · Zbl 1238.13005 [5] P. Gimenez, A. Simis, W. Vasconcelos and R. Villarreal, On complete monomial ideals, J. Commut. Algebra, to appear, arXiv:1310.7793. · Zbl 1353.13006 [6] Lejeune-Jalabert, M. and Teissier, B., Clôture intégrale des idéaux et équisingularité, Ann. Fac. Sci. Toulouse Math. (6)17(4) (2008) 781-859. · Zbl 1171.13005 [7] Reid, L., Roberts, L. G. and Vitulli, M. A., Some results on normal homogeneous ideals, Comm. Algebra31(9) (2003) 4485-4506. · Zbl 1021.13008 [8] Swanson, I. and Huneke, C., Integral Closure of Ideals, Rings and Modules, , Vol. 336 (Cambridge University Press, Cambridge, 2006). · Zbl 1117.13001
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