×

zbMATH — the first resource for mathematics

Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs. (English) Zbl 1243.14005
A sequence \(a_0, \dots, a_n\) of real numbers is called log-concave if \(a_{i-1}a_{i+1}\leq a_i^2\) for all \(i, 0<i<n\).
The aim of the paper is to answer the following question.
Let \(\chi_G(t)=a_nt^n-a_{n-1}t^{n-1}+\cdots +(-1)^n a_0\) be the chromatic polynomial of a graph \(G\). Is the sequence \(a_0, \dots, a_n\) log-concave? It is proved that for a matroid \(M\) which is representable over a field of characteristic zero the coefficients of the characteristic polynomial of \(M\) form a sign-alternating log-concave sequence of integers with no internal zeros.

MSC:
14B05 Singularities in algebraic geometry
05B35 Combinatorial aspects of matroids and geometric lattices
Software:
CSM-A
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Martin Aigner, Whitney numbers, Combinatorial geometries, Encyclopedia Math. Appl., vol. 29, Cambridge Univ. Press, Cambridge, 1987, pp. 139 – 160. · Zbl 0631.05015
[2] Paolo Aluffi, Computing characteristic classes of projective schemes, J. Symbolic Comput. 35 (2003), no. 1, 3 – 19. · Zbl 1074.14502 · doi:10.1016/S0747-7171(02)00089-5 · doi.org
[3] Francesco Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 71 – 89. · Zbl 0813.05007 · doi:10.1090/conm/178/01893 · doi.org
[4] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005
[5] David A. Cox, John Little, and Donal O’Shea, Using algebraic geometry, 2nd ed., Graduate Texts in Mathematics, vol. 185, Springer, New York, 2005. · Zbl 1079.13017
[6] Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. · Zbl 0753.57001
[7] Alexandru Dimca and Stefan Papadima, Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math. (2) 158 (2003), no. 2, 473 – 507. · Zbl 1068.32019 · doi:10.4007/annals.2003.158.473 · doi.org
[8] William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. · Zbl 0813.14039
[9] Terence Gaffney, Integral closure of modules and Whitney equisingularity, Invent. Math. 107 (1992), no. 2, 301 – 322. · Zbl 0807.32024 · doi:10.1007/BF01231892 · doi.org
[10] Terence Gaffney, Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), no. 2, 209 – 220. · Zbl 0846.32024 · doi:10.1007/s002220050022 · doi.org
[11] M. Gromov, Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1 – 38. · Zbl 0770.53042
[12] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995. A first course; Corrected reprint of the 1992 original. · Zbl 0779.14001
[13] Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017 – 1032. · Zbl 0304.14005
[14] A. P. Heron, Matroid polynomials, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), Inst. Math. Appl., Southend-on-Sea, 1972, pp. 164 – 202.
[15] Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006. · Zbl 1117.13001
[16] Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). · Zbl 0519.14002
[17] K. Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Preprint: arXiv:0904.3350v2. · Zbl 1270.14022
[18] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. · Zbl 0633.53002
[19] A. G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (1976), no. 1, 1 – 31 (French). · Zbl 0328.32007 · doi:10.1007/BF01389769 · doi.org
[20] Joseph P. S. Kung, The geometric approach to matroid theory, Gian-Carlo Rota on combinatorics, Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995, pp. 604 – 622.
[21] Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals.
[22] Robert Lazarsfeld and Mircea Mustaţă, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 5, 783 – 835 (English, with English and French summaries). · Zbl 1182.14004
[23] R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423 – 432. · Zbl 0311.14001 · doi:10.2307/1971080 · doi.org
[24] David Marker, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002. An introduction. · Zbl 1003.03034
[25] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145 – 158. · Zbl 0057.02601
[26] Andrei Okounkov, Brunn-Minkowski inequality for multiplicities, Invent. Math. 125 (1996), no. 3, 405 – 411. · Zbl 0893.52004 · doi:10.1007/s002220050081 · doi.org
[27] James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011. · Zbl 1254.05002
[28] Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167 – 189. · Zbl 0432.14016 · doi:10.1007/BF01392549 · doi.org
[29] Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. · Zbl 0757.55001
[30] Richard Randell, Morse theory, Milnor fibers and minimality of hyperplane arrangements, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2737 – 2743. · Zbl 1004.32010
[31] Ronald C. Read, An introduction to chromatic polynomials, J. Combinatorial Theory 4 (1968), 52 – 71. · Zbl 0173.26203
[32] D. Rees and R. Y. Sharp, On a theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc. (2) 18 (1978), no. 3, 449 – 463. · Zbl 0408.13009 · doi:10.1112/jlms/s2-18.3.449 · doi.org
[33] Gian-Carlo Rota, Combinatorial theory, old and new, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 229 – 233. · Zbl 0362.05044
[34] Pierre Samuel, La notion de multiplicité en algèbre et en géométrie algébrique, J. Math. Pures Appl. (9) 30 (1951), 159 – 205 (French). · Zbl 0044.02701
[35] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. · Zbl 0798.52001
[36] Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. · Zbl 1273.14004
[37] G. C. Shephard, Inequalities between mixed volumes of convex sets, Mathematika 7 (1960), 125 – 138. · Zbl 0108.35203 · doi:10.1112/S0025579300001674 · doi.org
[38] Richard P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Graph theory and its applications: East and West (Jinan, 1986) Ann. New York Acad. Sci., vol. 576, New York Acad. Sci., New York, 1989, pp. 500 – 535. · Zbl 0792.05008 · doi:10.1111/j.1749-6632.1989.tb16434.x · doi.org
[39] Richard P. Stanley, Foundations I and the development of algebraic combinatorics, Gian-Carlo Rota on combinatorics, Contemp. Mathematicians, Birkhäuser Boston, Boston, MA, 1995, pp. 105 – 107.
[40] Richard P. Stanley, Positivity problems and conjectures in algebraic combinatorics, Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 2000, pp. 295 – 319. · Zbl 0955.05111
[41] Richard P. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389 – 496. · Zbl 1136.52009
[42] Bernard Teissier, Cycles évanescents, sections planes et conditions de Whitney, Singularités à Cargèse (Rencontre Singularités Géom. Anal., Inst. Études Sci., Cargèse, 1972) Soc. Math. France, Paris, 1973, pp. 285 – 362. Astérisque, Nos. 7 et 8 (French). · Zbl 0295.14003
[43] David Eisenbud and Harold I. Levine, An algebraic formula for the degree of a \?^\infty map germ, Ann. of Math. (2) 106 (1977), no. 1, 19 – 44. With an appendix by Bernard Teissier, ”Sur une inégalité à la Minkowski pour les multiplicités”. · Zbl 0398.57020 · doi:10.2307/1971156 · doi.org
[44] Bernard Teissier, Du théorème de l’index de Hodge aux inégalités isopérimétriques, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 4, A287 – A289 (French, with English summary). · Zbl 0406.14011
[45] Bernard Teissier, Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 314 – 491 (French). · Zbl 0585.14008 · doi:10.1007/BFb0071291 · doi.org
[46] Ngô Viêt Trung, Positivity of mixed multiplicities, Math. Ann. 319 (2001), no. 1, 33 – 63. · Zbl 0979.13023 · doi:10.1007/PL00004429 · doi.org
[47] Ngo Viet Trung and Jugal Verma, Mixed multiplicities of ideals versus mixed volumes of polytopes, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4711 – 4727. · Zbl 1121.52027
[48] D. J. A. Welsh, Matroid theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L. M. S. Monographs, No. 8. · Zbl 0343.05002
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.