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Toward a theory of monomial preorders


Authors: Gregor Kemper, Ngo Viet Trung and Nguyen Thi Van Anh
Journal: Math. Comp.
MSC (2010): Primary 13P10; Secondary 14Qxx, 13H10
DOI: https://doi.org/10.1090/mcom/3289
Published electronically: December 28, 2017
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Abstract: In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone.


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  • [1] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478, https://doi.org/10.1006/jsco.1996.0125
  • [2] W. Bruns and A. Conca, Gröbner bases, initial ideals and initial algebras, In: L.L. Avramov et al. (Hrsg.), Homological Methods in Commutative Algebra, IPM Proceedings, Teheran, 2004.
  • [3] Maria Pia Cavaliere and Gianfranco Niesi, On Serre's conditions in the form ring of an ideal, J. Math. Kyoto Univ. 21 (1981), no. 3, 537-546. MR 629783, https://doi.org/10.1215/kjm/1250521977
  • [4] Giulio Caviglia, The pinched Veronese is Koszul, J. Algebraic Combin. 30 (2009), no. 4, 539-548. MR 2563140, https://doi.org/10.1007/s10801-009-0176-1
  • [5] Aldo Conca, Reduction numbers and initial ideals, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1015-1020. MR 1948090, https://doi.org/10.1090/S0002-9939-02-06607-8
  • [6] David Eisenbud, Commutative Algebra: With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960
  • [7] Günter Ewald and Masa-Nori Ishida, Completion of real fans and Zariski-Riemann spaces, Tohoku Math. J. (2) 58 (2006), no. 2, 189-218. MR 2248429
  • [8] Vesselin Gasharov, Noam Horwitz, and Irena Peeva, Hilbert functions over toric rings, Michigan Math. J. 57 (2008), 339-357. MR 2492457, https://doi.org/10.1307/mmj/1220879413
  • [9] Pedro D. González Pérez and Bernard Teissier, Toric geometry and the Semple-Nash modification, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 108 (2014), no. 1, 1-48. MR 3183106, https://doi.org/10.1007/s13398-012-0096-0
  • [10] Silvio Greco and Maria Grazia Marinari, Nagata's criterion and openness of loci for Gorenstein and complete intersection, Math. Z. 160 (1978), no. 3, 207-216. MR 0491741, https://doi.org/10.1007/BF01237034
  • [11] G.-M. Greuel and G. Pfister, Advances and improvements in the theory of standard bases and syzygies, Arch. Math. (Basel) 66 (1996), no. 2, 163-176. MR 1367159, https://doi.org/10.1007/BF01273348
  • [12] Gert-Martin Greuel and Gerhard Pfister, A Singular Introduction to Commutative Algebra, Second, extended edition, Springer, Berlin, 2008. MR 2363237
  • [13] A. Grothendieck, Éléments de géométrie algébrique IV (Seconde Partie), Publications Mathématiques 24, Institut des Hautes Études Scientifiques, 1965.
  • [14] Jürgen Herzog, Generators and relations of abelian semigroups and semigroup rings., Manuscripta Math. 3 (1970), 175-193. MR 0269762, https://doi.org/10.1007/BF01273309
  • [15] Craig Huneke, Lectures on local cohomology, Interactions between homotopy theory and algebra, Contemp. Math., vol. 436, Amer. Math. Soc., Providence, RI, 2007, pp. 51-99. MR 2355770, https://doi.org/10.1090/conm/436/08404
  • [16] Michael Kalkbrener and Bernd Sturmfels, Initial complexes of prime ideals, Adv. Math. 116 (1995), no. 2, 365-376. MR 1363769, https://doi.org/10.1006/aima.1995.1071
  • [17] Gregor Kemper and Ngo Viet Trung, Krull dimension and monomial orders, J. Algebra 399 (2014), 782-800. MR 3144612, https://doi.org/10.1016/j.jalgebra.2013.10.005
  • [18] Heinz Kredel and Volker Weispfenning, Computing dimension and independent sets for polynomial ideals, J. Symbolic Comput. 6 (1988), no. 2-3, 231-247. Computational aspects of commutative algebra. MR 988415, https://doi.org/10.1016/S0747-7171(88)80045-2
  • [19] Martin Kreuzer and Lorenzo Robbiano, Computational Commutative Algebra. 2, Springer-Verlag, Berlin, 2005. MR 2159476
  • [20] Ferdinando Mora, An algorithm to compute the equations of tangent cones, Computer algebra (Marseille, 1982) Lecture Notes in Comput. Sci., vol. 144, Springer, Berlin-New York, 1982, pp. 158-165. MR 680065
  • [21] Teo Mora, Solving Polynomial Equation Systems. II: Macaulay's Paradigm and Gröbner Technology, Encyclopedia of Mathematics and its Applications, vol. 99, Cambridge University Press, Cambridge, 2005. MR 2164357
  • [22] Teo Mora and Lorenzo Robbiano, The Gröbner fan of an ideal, J. Symbolic Comput. 6 (1988), no. 2-3, 183-208. Computational aspects of commutative algebra. MR 988412, https://doi.org/10.1016/S0747-7171(88)80042-7
  • [23] Edward Mosteig and Moss Sweedler, Valuations and filtrations, J. Symbolic Comput. 34 (2002), no. 5, 399-435. MR 1937467, https://doi.org/10.1006/jsco.2002.0565
  • [24] Lorenzo Robbiano, Term orderings on the polynomial ring, EUROCAL '85, Vol. 2 (Linz, 1985) Lecture Notes in Comput. Sci., vol. 204, Springer, Berlin, 1985, pp. 513-517. MR 826583, https://doi.org/10.1007/3-540-15984-3_321
  • [25] Lorenzo Robbiano, On the theory of graded structures, J. Symbolic Comput. 2 (1986), no. 2, 139-170. MR 849048, https://doi.org/10.1016/S0747-7171(86)80019-0
  • [26] Enrico Sbarra, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra 29 (2001), no. 12, 5383-5409. MR 1872238, https://doi.org/10.1081/AGB-100107934
  • [27] Adam S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36 (2004), no. 4, 519-526. MR 2069015, https://doi.org/10.1112/S0024609303003060
  • [28] Toshio Sumi, Mitsuhiro Miyazaki, and Toshio Sakata, Typical ranks for 3-tensors, nonsingular bilinear maps and determinantal ideals, J. Algebra 471 (2017), 409-453. MR 3569191, https://doi.org/10.1016/j.jalgebra.2016.09.028
  • [29] Ngô Viêt Trung, Constructive characterization of the reduction numbers, Compositio Math. 137 (2003), no. 1, 99-113. MR 1981939, https://doi.org/10.1023/A:1013940219415
  • [30] Matteo Varbaro, Gröbner deformations, connectedness and cohomological dimension, J. Algebra 322 (2009), no. 7, 2492-2507. MR 2553691, https://doi.org/10.1016/j.jalgebra.2009.01.018
  • [31] Wolmer V. Vasconcelos, Cohomological degrees of graded modules, Six Lectures on Commutative Algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 345-392. MR 1648669

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Additional Information

Gregor Kemper
Affiliation: Technische Universiät München, Zentrum Mathematik - M11, Boltzmannstr. 3, 85748 Garching, Germany
Email: kemper@ma.tum.de

Ngo Viet Trung
Affiliation: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam
Email: nvtrung@math.ac.vn

Nguyen Thi Van Anh
Affiliation: University of Osnabrück, Institut für Mathematik, Albrechtstr. 28 A, 49076 Osnabrück, Germany
Email: ngthvanh@gmail.com

DOI: https://doi.org/10.1090/mcom/3289
Keywords: Monomial order, monomial preorder, weight order, leading ideal, standard basis, flat deformation, dimension, descent of properties, regular locus, normal locus, Cohen-Macaulay locus, graded invariants, toric ring
Received by editor(s): September 16, 2016
Received by editor(s) in revised form: April 12, 2017
Published electronically: December 28, 2017
Additional Notes: The second author was supported by the Vietnam National Foundation for Science and Technology Development under grant number 101.04-2017.19 and the Project VAST.HTQT.NHAT.1/16-18. A large part of the paper was completed during a long term visit of the second author to Vietnam Institute for Advanced Study in Mathematics.
Article copyright: © Copyright 2017 American Mathematical Society

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