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Forum Mathematicum

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Regularity of symbolic powers and arboricity of matroids

Nguyên Công Minh
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  • Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam
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/ Trân Nam Trung
Published Online: 2018-10-30 | DOI: https://doi.org/10.1515/forum-2017-0243

Abstract

Let Δ be a matroid complex. In this paper, we explicitly compute the regularity of all the symbolic powers of its Stanley–Reisner ideal in terms of combinatorial data of Δ. In order to do that, we provide a sharp bound between the arboricity of Δ and the circumference of its dual Δ*.

Keywords: Matroid; arboricity; circumference; Stanley–Reisner ideal; Castelnuovo–Mumford regularity

MSC 2010: 13D45; 05E40; 05E45

References

  • [1]

    N. Alon, C. McDiarmid and B. Reed, Star arboricity, Combinatorica 12 (1992), no. 4, 375–380. CrossrefGoogle Scholar

  • [2]

    S. Beyarslan, H. T. Hà and T. N. Trung, Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095. CrossrefGoogle Scholar

  • [3]

    M. Chardin, Powers of ideals and the cohomology of stalks and fibers of morphisms, Algebra Number Theory 7 (2013), no. 1, 1–18. Web of ScienceCrossrefGoogle Scholar

  • [4]

    S. D. Cutkosky, Irrational asymptotic behaviour of Castelnuovo–Mumford regularity, J. Reine Angew. Math. 522 (2000), 93–103. Google Scholar

  • [5]

    S. D. Cutkosky, J. Herzog and N. V. Trung, Asymptotic behaviour of the Castelnuovo–Mumford regularity, Compos. Math. 118 (1999), no. 3, 243–261. CrossrefGoogle Scholar

  • [6]

    H. Dao, A. De Stefani, E. Grifo, C. Huneke and L. Núñez Betancourt, Symbolic powers of ideals, Singularities and Foliations. Geometry, Topology and Applications, Springer Proc. Math. Stat. 222, Springer, Cham (2018), 387–432. Google Scholar

  • [7]

    A. M. Dean, J. P. Hutchinson and E. R. Scheinerman, On the thickness and arboricity of a graph, J. Combin. Theory Ser. B 52 (1991), no. 1, 147–151. CrossrefGoogle Scholar

  • [8]

    J. Edmonds, Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 67–72. CrossrefGoogle Scholar

  • [9]

    D. Eisenbud, The Geometry of Syzygies, Grad. Texts in Math. 229, Springer, New York, 2005. Google Scholar

  • [10]

    D. Eisenbud and J. Harris, Powers of ideals and fibers of morphisms, Math. Res. Lett. 17 (2010), no. 2, 267–273. CrossrefGoogle Scholar

  • [11]

    D. Eisenbud and B. Ulrich, Notes on regularity stabilization, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1221–1232. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    J. Herzog, T. Hibi and N. V. Trung, Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), no. 1, 304–322. CrossrefWeb of ScienceGoogle Scholar

  • [13]

    J. Herzog, L. T. Hoa and N. V. Trung, Asymptotic linear bounds for the Castelnuovo–Mumford regularity, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1793–1809. CrossrefGoogle Scholar

  • [14]

    T. Hibi, Buchsbaum complexes with linear resolutions, J. Algebra 179 (1996), no. 1, 127–136. CrossrefGoogle Scholar

  • [15]

    L. T. Hoa and T. N. Trung, Partial Castelnuovo–Mumford regularities of sums and intersections of powers of monomial ideals, Math. Proc. Cambridge Philos. Soc. 149 (2010), no. 2, 229–246. CrossrefWeb of ScienceGoogle Scholar

  • [16]

    L. T. Hoa and T. N. Trung, Castelnuovo–Mumford regularity of symbolic powers of two-dimensional square-free monomial ideals, J. Commut. Algebra 8 (2016), no. 1, 77–88. Web of ScienceCrossrefGoogle Scholar

  • [17]

    V. Kodiyalam, Asymptotic behaviour of Castelnuovo–Mumford regularity, Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411. CrossrefGoogle Scholar

  • [18]

    N. C. Minh, N. Terai and P. T. Thuy, Level property of ordinary and symbolic powers of Stanley–Reisner ideals, preprint (2018).

  • [19]

    N. C. Minh and N. V. Trung, Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals, Adv. Math. 226 (2011), no. 2, 1285–1306. Web of ScienceCrossrefGoogle Scholar

  • [20]

    C. S. J. A. Nash-Williams, Decomposition of finite graphs into forests, J. Lond. Math. Soc. 39 (1964), 1–12. Google Scholar

  • [21]

    J. G. Oxley, Matroid Theory, Clarendon Press, New York, 1992. Google Scholar

  • [22]

    P. D. Seymour, A note on list arboricity, J. Combin. Theory Ser. B 72 (1998), no. 1, 150–151. CrossrefGoogle Scholar

  • [23]

    R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progr. Math. 41, Birkhäuser, Boston, 1996. Google Scholar

  • [24]

    Y. Takayama, Combinatorial characterizations of generalized Cohen–Macaulay monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 48(96) (2005), no. 3, 327–344. Google Scholar

  • [25]

    N. Terai and N. V. Trung, Cohen–Macaulayness of large powers of Stanley–Reisner ideals, Adv. Math. 229 (2012), no. 2, 711–730. Web of ScienceCrossrefGoogle Scholar

  • [26]

    N. V. Trung and H.-J. Wang, On the asymptotic linearity of Castelnuovo–Mumford regularity, J. Pure Appl. Algebra 201 (2005), no. 1–3, 42–48. CrossrefGoogle Scholar

  • [27]

    M. Varbaro, Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2357–2366. Web of ScienceGoogle Scholar

  • [28]

    H. Whitney, On the abstract properties of linear dependence, Amer. J. Math. 57 (1935), no. 3, 509–533. CrossrefGoogle Scholar

About the article


Received: 2017-11-21

Revised: 2018-09-28

Published Online: 2018-10-30


This paper was done while the first author was visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for the hospitality and financial support, and he also thanks the Vietnam National Foundation for Science and Technology Development (NAFOSTED) for its support under grant number 101.01-2016.21.


Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0243.

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