Toward a theory of monomial preorders

Kemper, Gregor; Trung, Ngo; Van Anh, Nguyen Thi

doi:10.1090/mcom/3289

Toward a theory of monomial preorders
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by Gregor Kemper, Ngo Viet Trung and Nguyen Thi Van Anh;

Math. Comp. 87 (2018), 2513-2537

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.

References

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, DOI 10.1006/jsco.1996.0125
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.
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, DOI 10.1215/kjm/1250521977
Giulio Caviglia, The pinched Veronese is Koszul, J. Algebraic Combin. 30 (2009), no. 4, 539–548. MR 2563140, DOI 10.1007/s10801-009-0176-1
Aldo Conca, Reduction numbers and initial ideals, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1015–1020. MR 1948090, DOI 10.1090/S0002-9939-02-06607-8
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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
Vesselin Gasharov, Noam Horwitz, and Irena Peeva, Hilbert functions over toric rings, Michigan Math. J. 57 (2008), 339–357. Special volume in honor of Melvin Hochster. MR 2492457, DOI 10.1307/mmj/1220879413
Pedro D. González Pérez and Bernard Teissier, Toric geometry and the Semple-Nash modification, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 108 (2014), no. 1, 1–48. MR 3183106, DOI 10.1007/s13398-012-0096-0
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 491741, DOI 10.1007/BF01237034
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, DOI 10.1007/BF01273348
Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Second, extended edition, Springer, Berlin, 2008. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann; With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2363237
A. Grothendieck, Éléments de géométrie algébrique IV (Seconde Partie), Publications Mathématiques 24, Institut des Hautes Études Scientifiques, 1965.
Jürgen Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175–193. MR 269762, DOI 10.1007/BF01273309
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. Appendix 1 by Amelia Taylor. MR 2355770, DOI 10.1090/conm/436/08404
Michael Kalkbrener and Bernd Sturmfels, Initial complexes of prime ideals, Adv. Math. 116 (1995), no. 2, 365–376. MR 1363769, DOI 10.1006/aima.1995.1071
Gregor Kemper and Ngo Viet Trung, Krull dimension and monomial orders, J. Algebra 399 (2014), 782–800. MR 3144612, DOI 10.1016/j.jalgebra.2013.10.005
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, DOI 10.1016/S0747-7171(88)80045-2
Martin Kreuzer and Lorenzo Robbiano, Computational commutative algebra. 2, Springer-Verlag, Berlin, 2005. MR 2159476
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
Teo Mora, Solving polynomial equation systems. II, Encyclopedia of Mathematics and its Applications, vol. 99, Cambridge University Press, Cambridge, 2005. Macaulay’s paradigm and Gröbner technology. MR 2164357, DOI 10.1017/CBO9781107340954
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, DOI 10.1016/S0747-7171(88)80042-7
Edward Mosteig and Moss Sweedler, Valuations and filtrations, J. Symbolic Comput. 34 (2002), no. 5, 399–435. MR 1937467, DOI 10.1006/jsco.2002.0565
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, DOI 10.1007/3-540-15984-3_{3}21
Lorenzo Robbiano, On the theory of graded structures, J. Symbolic Comput. 2 (1986), no. 2, 139–170. MR 849048, DOI 10.1016/S0747-7171(86)80019-0
Enrico Sbarra, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra 29 (2001), no. 12, 5383–5409. MR 1872238, DOI 10.1081/AGB-100107934
Adam S. Sikora, Topology on the spaces of orderings of groups, Bull. London Math. Soc. 36 (2004), no. 4, 519–526. MR 2069015, DOI 10.1112/S0024609303003060
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, DOI 10.1016/j.jalgebra.2016.09.028
Ngô Viêt Trung, Constructive characterization of the reduction numbers, Compositio Math. 137 (2003), no. 1, 99–113. MR 1981939, DOI 10.1023/A:1013940219415
Matteo Varbaro, Gröbner deformations, connectedness and cohomological dimension, J. Algebra 322 (2009), no. 7, 2492–2507. MR 2553691, DOI 10.1016/j.jalgebra.2009.01.018
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