×

zbMATH — the first resource for mathematics

The entropic discriminant. (English) Zbl 1284.15006
Summary: The entropic discriminant is a nonnegative polynomial associated to a matrix. It arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry. It describes the complex branch locus of the polar map of a real hyperplane arrangement, and it vanishes when the equations defining the analytic center of a linear program have a complex double root. We study the geometry of the entropic discriminant, and we express its degree in terms of the characteristic polynomial of the underlying matroid. Singularities of reciprocal linear spaces play a key role. In the corank-one case, the entropic discriminant admits a sum of squares representation derived from the discriminant of a characteristic polynomial of a symmetric matrix.

MSC:
15A21 Canonical forms, reductions, classification
05B35 Combinatorial aspects of matroids and geometric lattices
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
14M12 Determinantal varieties
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alexandersson, P.; Shapiro, B., Discriminants, symmetrized graph monomials and sums of squares, Experiment. Math., 21, 353-361, (2012) · Zbl 1259.13012
[2] Borchardt, C. W., Neue eigenschaft der gleichung, mit deren Hülfe man die seculären störungen der planeten bestimmt, J. Reine Angew. Math., 30, 38-45, (1846)
[3] de Loera, J. A.; Sturmfels, B.; Vinzant, C., The central curve in linear programming, Found. Comput. Math., 12, 509-540, (2012) · Zbl 1254.90108
[4] Dimca, A.; Papadima, S., Hypersurface complements, Milnor fibers and higher homotopy groups of arrangments, Ann. of Math., 158, 473-507, (2003) · Zbl 1068.32019
[5] Dolgachev, I. V., Polar Cremona transformations, Michigan Math. J., 48, 191-202, (2000) · Zbl 1080.14511
[6] Dolgachev, I. V., Classical algebraic geometry: A modern view, (2012), Cambridge Univ. Press · Zbl 1252.14001
[7] Domokos, M., The discriminant of symmetric matrices as a sum of squares and the orthogonal group, Comm. Pure Appl. Math., 64, 443-465, (2011) · Zbl 1219.15008
[8] Fischer, G., (Plane Algebraic Curves, Student Mathematical Library, vol. 15, (2001), American Mathematical Society Providence, RI)
[9] Flenner, H.; O’Carroll, L.; Vogel, W., Joins and intersections, (1999), Springer Verlag New York · Zbl 0939.14003
[10] Greene, C.; Zaslavsky, T., On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs, Trans. Amer. Math. Soc., 280, 97-126, (1983) · Zbl 0539.05024
[11] Hillar, C.; Wibisono, A., Maximum entropy distributions on graphs
[12] Huh, J., Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc., 25, 907-927, (2012) · Zbl 1243.14005
[13] Huh, J.; Katz, E., Log-concavity of characteristic polynomials and the Bergman Fan of matroids, Math. Ann., 354, 1103-1116, (2012) · Zbl 1258.05021
[14] Ilyushechkin, N. V., The discriminant of the characteristic polynomial of a normal matrix, Mat. Zametki, 51, 16-23, (1992) · Zbl 0796.15009
[15] Lax, P. D., On the discriminant of real symmetric matrices, Comm. Pure Appl. Math., 51, 1387-1396, (1998) · Zbl 0933.15013
[16] Muir, T., A treatise on the theory of determinants, (1960), Dover Publications Inc. New York, Revised and enlarged by William H. Metzler
[17] Nagata, M., On the purity of branch loci in regular local rings, Illinois J. Math., 3, 328-333, (1959) · Zbl 0115.26201
[18] Newell, M. J., On identities associated with a discriminant, Proc. Edinb. Math. Soc., 18, 287-291, (1972-1973) · Zbl 0269.15005
[19] Oxley, J., Matroid theory, (1992), Oxford University Press · Zbl 0784.05002
[20] Proudfoot, N.; Speyer, D., A broken circuit ring, Beitr. Algebra Geom., 47, 161-166, (2006) · Zbl 1095.13024
[21] Sanyal, R., On the derivative cones of polyhedral cones, Adv. Geom., 13, 315-321, (2013) · Zbl 1264.90135
[22] Schenck, H.; Tohaˇneanu, Ş., The orlik-terao algebra and 2-formality, Math. Res. Lett., 16, 171-182, (2009) · Zbl 1171.52011
[23] Sottile, F., (Real Solutions to Equations from Geometry, University Lecture Series, (2011), American Mathematical Society Providence, Rhode Island) · Zbl 1233.14001
[24] Stanley, R. P., An introduction to hyperplane arrangements, (Geometric Combinatorics, IAS/Park City Math. Ser., vol. 13, (2007), American Mathematical Society Providence, Rhode Island), 389-496 · Zbl 1136.52009
[25] Sturmfels, B., (Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, vol. 97, (2002), American Mathematical Society Providence, Rhode Island) · Zbl 1101.13040
[26] Tatakis, C.; Thoma, A., On the universal Gröbner bases of toric ideals of graphs, J. Combin. Theory Ser. A, 118, 1540-1548, (2011) · Zbl 1232.05094
[27] Varchenko, A., Critical points of the product of powers of linear functions and families of bases of singular vectors, Compos. Math., 97, 385-401, (1995) · Zbl 0842.17044
[28] (White, N., Combinatorial Geometries, Encyclopedia of Mathematics and its Applications, vol. 29, (1987), Cambridge University Press Cambridge)
[29] J. Yu, D. Yuster, Representing tropical linear spaces by circuits, in: Proceedings of FPSAC, 2007.
[30] Zaslavsky, T., Chromatic invariants of signed graphs, Discrete Math., 42, 287-312, (1982) · Zbl 0498.05030
[31] Zaslavsky, T., Signed graphs, Discrete Appl. Math., 4, 47-74, (1982) · Zbl 0476.05080
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.