×

zbMATH — the first resource for mathematics

Primary components of codimension two lattice basis ideals. (English) Zbl 1387.13048
A codimension two lattice basis ideal is a binomial ideal generated by two binomials whose exponent vectors are linearly independent. The present paper characterizes the binomial primary decomposition of such an ideal. It thereby, in this special case, finishes a theory initiated in [S. Hoşten and J. Shapiro, J. Symb. Comput. 29, No. 4–5, 625–639 (2000; Zbl 0968.13003)] where the associated primes have been determined, and [A. Dickenstein et al., Math. Z. 264, No. 4, 745–763 (2010; Zbl 1190.13017)] where the so-called toral primary components have been characterized. Describing the remaining Andean primary components is reduced to certain graph theoretic problems which form the core of this paper.
As an application, the authors compute the set of parameters for which a bivariate Horn system of hypergeometric differential equations is holonomic.
MSC:
13F99 Arithmetic rings and other special commutative rings
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
33C70 Other hypergeometric functions and integrals in several variables
20M25 Semigroup rings, multiplicative semigroups of rings
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abbott, J., Bigatti, A.M., Lagorio, G.: CoCoA-5: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.it · Zbl 1440.62394
[2] Berkesch-Zamaere, C., Matusevich, L.F., Walther, U.: Torus equivariant D-modules and hypergeometric systems. Preprint arXiv:1308.5901 (2013) · Zbl 0968.13003
[3] Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-6 — a computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2012) · Zbl 1190.13017
[4] Dickenstein, A.; Matusevich, L.F.; Miller, E., Combinatorics of binomial primary decomposition, Math. Z., 264, 745-763, (2010) · Zbl 1190.13017
[5] Dickenstein, A.; Matusevich, L.F.; Miller, E., Binomial D-modules, Duke Math. J., 151, 385-429, (2010) · Zbl 1205.13031
[6] Dickenstein, A.; Matusevich, L.F.; Sadikov, T., Bivariate hypergeometric D-modules, Adv. Math., 196, 78-123, (2005) · Zbl 1089.33009
[7] Eisenbud, D.; Sturmfels, B., Binomial ideals, Duke Math. J., 84, 1-45, (1996) · Zbl 0873.13021
[8] Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
[9] Hoşten, S.; Shapiro, J., Primary decomposition of lattice basis ideals, J. Symbolic Comput., 29, 625-639, (2000) · Zbl 0968.13003
[10] Kahle, T., Decompositions of binomial ideals, Ann. Inst. Statist. Math., 62, 727-745, (2010) · Zbl 1440.62394
[11] Mayr, E.; Meyer, A., The complexity of the word problem for commutative semigroups and polynomial ideals, Adv. Math., 46, 305-329, (1982) · Zbl 0506.03007
[12] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, Vol. 227. Springer-Verlag, New York (2005) · Zbl 0968.13003
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.