Ehrhart tensor polynomials. (English) Zbl 1377.05006
Summary: The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by M. Ludwig and L. Silverstein [Adv. Math. 319, 76–110 (2017; Zbl 1390.52023)]. We initiate a study of their coefficients. In the vector and matrix cases, we give Pick-type formulas in terms of triangulations of a lattice polygon. As our main tool, we introduce \(h^r\)-tensor polynomials, extending the notion of the Ehrhart \(h^\ast\)-polynomial, and, for matrices, investigate their coefficients for positive semidefiniteness. In contrast to the usual \(h^\ast\)-polynomial, the coefficients are in general not monotone with respect to inclusion. Nevertheless, we are able to prove positive semidefiniteness in dimension two. Based on computational results, we conjecture positive semidefiniteness of the coefficients in higher dimensions. Furthermore, we generalize Hibi’s palindromic theorem for reflexive polytopes to \(h^r\)-tensor polynomials and discuss possible future research directions.
MSC:
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
| 05A15 | Exact enumeration problems, generating functions |
| 15A45 | Miscellaneous inequalities involving matrices |
| 15A69 | Multilinear algebra, tensor calculus |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 52B45 | Dissections and valuations (Hilbert’s third problem, etc.) |
Keywords:
Ehrhart tensor polynomial; \(h^r\)-tensor polynomial; Pick formula; positive semidefinite coefficients; half-open polytopesCitations:
Zbl 1390.52023Software:
LattEReferences:
| [1] | Alesker, S., Integrals of smooth and analytic functions over Minkowski’s sums of convex sets, (Convex Geometric Analysis. Convex Geometric Analysis, Berkeley, CA, 1996. Convex Geometric Analysis. Convex Geometric Analysis, Berkeley, CA, 1996, Math. Sci. Res. Inst. Publ., vol. 34 (1999), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 1-15 · Zbl 0933.46022 |
| [2] | Assarf, B.; Gawrilow, E.; Herr, K.; Joswig, M.; Lorenz, B.; Paffenholz, A.; Rehn, T., Computing convex hulls and counting integer points with, Math. Program. Comput., 9, 1-38 (2017) · Zbl 1370.90009 |
| [3] | Athanasiadis, C. A., \(h^\ast \)-Vectors, Eulerian polynomials and stable polytopes of graphs, Electron. J. Combin., 11 (2004/06), Research Paper #R6 · Zbl 1068.52016 |
| [4] | Barvinok, A. I., A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper. Res., 19, 769-779 (1994) · Zbl 0821.90085 |
| [5] | Beck, M., Multidimensional Ehrhart reciprocity, J. Combin. Theory Ser. A, 97, 187-197 (2002) · Zbl 1005.52006 |
| [6] | Beck, M.; Robins, S., Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, Undergraduate Texts in Mathematics (2015), Springer: Springer New York, electronically available at · Zbl 1339.52002 |
| [7] | Berenstein, A. D.; Zelevinsky, A. V., Tensor product multiplicities and convex polytopes in partition space, J. Geom. Phys., 5, 453-472 (1988) · Zbl 0712.17006 |
| [8] | Bihan, F., Irrational mixed decomposition and sharp fewnomial bounds for tropical polynomial systems, Discrete Comput. Geom., 55, 907-933 (2016) · Zbl 1375.14210 |
| [9] | K.J. Böröczky, M. Ludwig, Minkowski valuations on lattice polytopes, J. Eur. Math. Soc. (JEMS), in press.; K.J. Böröczky, M. Ludwig, Minkowski valuations on lattice polytopes, J. Eur. Math. Soc. (JEMS), in press. · Zbl 1506.52016 |
| [10] | Böröczky, K. J.; Ludwig, M., Valuations on lattice polytopes, (Kinderlen, M.; Vedel Jensen, E., Tensor Valuations and Their Applications in Stochastic Geometry and Imaging (2017)), 213-234 · Zbl 1376.52022 |
| [11] | Chapoton, F., q-Analogues of Ehrhart polynomials, Proc. Edinb. Math. Soc. (2), 59, 339-358 (2016) · Zbl 1343.52012 |
| [12] | De Loera, J. A.; Hemmecke, R.; Tauzer, J.; Yoshida, R., Effective lattice point counting in rational convex polytopes, J. Symbolic Comput., 38, 1273-1302 (2004) · Zbl 1137.52303 |
| [13] | Ehrhart, E., Sur les polyèdres rationnels homothétiques à \(n\) dimensions, C. R. Acad. Sci. Paris, 254, 616-618 (1962) · Zbl 0100.27601 |
| [14] | Ehrhart, E., Sur un problème de géométrie diophantienne linéaire, J. Reine Angew. Math., 227, 25-49 (1967) · Zbl 0155.37503 |
| [15] | Gawrilow, E.; Joswig, M., : a framework for analyzing convex polytopes, (Kalai, G.; Ziegler, G. M., Polytopes — Combinatorics and Computation (2000), Birkhäuser), 43-74 · Zbl 0960.68182 |
| [16] | Gruber, P. M., Convex and Discrete Geometry (2007), Springer: Springer Berlin · Zbl 1139.52001 |
| [17] | Haase, C.; Juhnke-Kubitzke, M.; Sanyal, R.; Theobald, T., Mixed Ehrhart polynomials, Electron. J. Combin., 24 (2017), Research Paper #P1.10 · Zbl 1360.52008 |
| [18] | Henk, M.; Tagami, M., Lower bounds on the coefficients of Ehrhart polynomials, European J. Combin., 30, 70-83 (2009) · Zbl 1158.52014 |
| [19] | Hibi, T., Ehrhart polynomials of convex polytopes, h-vectors of simplicial complexes, and nonsingular projective toric varieties, (Discrete and Computational Geometry. Discrete and Computational Geometry, New Brunswick, NJ, 1989/1990. Discrete and Computational Geometry. Discrete and Computational Geometry, New Brunswick, NJ, 1989/1990, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6 (1991), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 165-177 · Zbl 0755.05098 |
| [20] | Hibi, T., A lower bound theorem for Ehrhart polynomials of convex polytopes, Adv. Math., 105, 162-165 (1994) · Zbl 0807.52011 |
| [21] | K. Jochemko, R. Sanyal, Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem, J. Eur. Math. Soc. (JEMS), in press.; K. Jochemko, R. Sanyal, Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem, J. Eur. Math. Soc. (JEMS), in press. · Zbl 1375.52015 |
| [22] | Jochemko, K.; Sanyal, R., Combinatorial mixed valuations, Adv. Math., 319, 630-653 (2017) · Zbl 1375.52015 |
| [23] | Köppe, M.; Verdoolaege, S., Computing parametric rational generating functions with a primal Barvinok algorithm, Electron. J. Combin., 15 (2008), Research Paper #R16 · Zbl 1180.52014 |
| [24] | Lepelley, D.; Louichi, A.; Smaoui, H., On Ehrhart polynomials and probability calculations in voting theory, Soc. Choice Welf., 30, 363-383 (2008) · Zbl 1149.91028 |
| [25] | Liu, H.; Zong, C., On the classification of convex lattice polytopes, Adv. Geom., 11, 711-729 (2011) · Zbl 1243.52011 |
| [26] | Ludwig, M.; Silverstein, L., Tensor valuations on lattice polytopes, Adv. Math., 319, 76-110 (2017) · Zbl 1390.52023 |
| [27] | Macdonald, I. G., Polynomials associated with finite cell complexes, J. Lond. Math. Soc. (2), 4, 181-192 (1971) · Zbl 0216.45205 |
| [28] | McMullen, P., Valuations and Euler-type relations on certain classes of convex polytopes, Proc. Lond. Math. Soc. (3), 35, 113-135 (1977) · Zbl 0353.52001 |
| [29] | Miller, E.; Sturmfels, B., Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227 (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1090.13001 |
| [30] | Pick, G., Geometrisches zur Zahlenlehre, 311-319 (1899), Naturwiss. Zeitschr. Lotus: Naturwiss. Zeitschr. Lotus Prag · JFM 33.0216.01 |
| [31] | Pukhlikov, A. V.; Khovanskiĭ, A. G., Finitely additive measures of virtual polyhedra, Algebra i Analiz, 4, 161-185 (1992) · Zbl 0778.52006 |
| [32] | Schneider, R., Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and Its Applications, vol. 151 (2014), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1287.52001 |
| [33] | Schulz, C., A Discrete Moment Problem on Lattice Polytopes (2015), Freie Universität Berlin: Freie Universität Berlin Germany, Master’s thesis |
| [34] | Silverstein, L., Tensor Valuations on Lattice Polytopes (2017), Technische Universität Wien: Technische Universität Wien Austria, PhD thesis · Zbl 1390.52023 |
| [35] | Stanley, R. P., Combinatorial reciprocity theorems, Adv. Math., 14, 194-253 (1974) · Zbl 0294.05006 |
| [36] | Stanley, R. P., Decompositions of rational convex polytopes, Ann. Discrete Math., 6, 333-342 (1980) · Zbl 0812.52012 |
| [37] | Stanley, R. P., A monotonicity property of \(h\)-vectors and \(h^\ast \)-vectors, European J. Combin., 14, 251-258 (1993) · Zbl 0799.52008 |
| [38] | Stapledon, A., Inequalities and Ehrhart \(δ\)-vectors, Trans. Amer. Math. Soc., 361, 5615-5626 (2009) · Zbl 1181.52024 |
| [39] | Stapledon, A., Equivariant Ehrhart theory, Adv. Math., 226, 3622-3654 (2011) · Zbl 1218.52014 |
| [40] | Stapledon, A., Additive number theory and inequalities in Ehrhart theory, Int. Math. Res. Not. IMRN, 1497-1540 (2016) · Zbl 1342.52017 |
| [41] | Ziegler, G. M., Lectures on Polytopes (1995), Springer-Verlag: Springer-Verlag New York, revised edition, 1998, “Updates, corrections, and more” at · Zbl 0823.52002 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
