Ehrhart polynomial roots of reflexive polytopes. (English) Zbl 1497.52019
Summary: Recent work has focused on the roots \(z\in\mathbb{C}\) of the Ehrhart polynomial of a lattice polytope \(P\). The case when \(\operatorname{Re}(z)=-1/2\) is of particular interest: these polytopes satisfy Golyshev’s “canonical line hypothesis”. We characterise such polytopes when \(\dim(P)\leq 7\). We also consider the “half-strip condition”, where all roots \(z\) satisfy \(-\dim(P)/2\leq\operatorname{Re}{z}\leq \dim(P)/2-1\), and show that this holds for any reflexive polytope with \(\dim(P)\leq 5\). We give an example of a \(10\)-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi-Shibata in dimension \(34\).
MSC:
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 05A15 | Exact enumeration problems, generating functions |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
References:
| [1] | Victor V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom., 3(3):493-535, 1994. · Zbl 0829.14023 |
| [2] | M. Beck, J. A. De Loera, M. Develin, J. Pfeifle, and R. P. Stanley. Coefficients and roots of Ehrhart polynomials. In Integer points in polyhedra—geometry, number theory, algebra, optimization, volume 374 of Contemp. Math., pages 15-36. Amer. Math. Soc., Providence, RI, 2005. · Zbl 1153.52300 |
| [3] | Christian Bey, Martin Henk, and J¨org M. Wills. Notes on the roots of Ehrhart polynomials. Discrete Comput. Geom., 38(1):81-98, 2007. · Zbl 1126.52012 |
| [4] | A. A. Borisov and L. A. Borisov.Singular toric Fano three-folds.Mat. Sb., 183(2):134-141, 1992. text in Russian. English transl.: Russian Acad. Sci. Sb. Math., 75 (1993), 277-283. · Zbl 0786.14028 |
| [5] | Benjamin Braun. Norm bounds for Ehrhart polynomial roots. Discrete Comput. Geom., 39(1-3):191-193, 2008. · Zbl 1141.52017 |
| [6] | Benjamin Braun and Mike Develin. Ehrhart polynomial roots and Stanley’s nonnegativity theorem. In Integer points in polyhedra—geometry, number theory, representation theory, algebra, optimization, statistics, volume 452 of Contemp. Math., pages 67-78. Amer. Math. Soc., Providence, RI, 2008. · Zbl 1161.52310 |
| [7] | Weronika Buczy´nska. Fake weighted projective spaces.arXiv:0805.1211v1, May 2008. |
| [8] | Heinke Conrads. Weighted projective spaces and reflexive simplices. Manuscripta Math., 107(2):215-227, 2002. · Zbl 1013.52009 |
| [9] | David A. Cox and Sheldon Katz. Mirror symmetry and algebraic geometry, volume 68 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1999. · Zbl 0951.14026 |
| [10] | Eug‘ene Ehrhart. Sur les poly‘edres homoth´etiques bord´es ‘a n dimensions. C. R. Acad. Sci. Paris, 254:988-990, 1962. · Zbl 0100.27602 |
| [11] | Eug‘ene Ehrhart. Sur un probl‘eme de g´eom´etrie diophantienne lin´eaire. II. Syst‘emes diophantiens lin´eaires. J. Reine Angew. Math., 227:25-49, 1967. · Zbl 0155.37503 |
| [12] | V. V. Golyshev. On the canonical strip. Uspekhi Mat. Nauk, 64(1(385)):139-140, 2009. · Zbl 1172.14330 |
| [13] | G´abor Heged¨us and Alexander M. Kasprzyk. Roots of Ehrhart polynomials of smooth Fano polytopes. Discrete Comput. Geom., 46(3):488-499, 2011. · Zbl 1230.52026 |
| [14] | Takayuki Hibi. Ehrhart polynomials of convex polytopes, h-vectors of simplicial complexes, and nonsingular projective toric varieties. In Discrete and computational geometry (New Brunswick, NJ, 1989/1990), volume 6 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci., pages 165-177. Amer. Math. Soc., Providence, RI, 1991. the electronic journal of combinatorics 26(1) (2019), #P1.3826 · Zbl 0755.05098 |
| [15] | Takayuki Hibi. A lower bound theorem for Ehrhart polynomials of convex polytopes. Adv. Math., 105(2):162-165, 1994. · Zbl 0807.52011 |
| [16] | Akihiro Higashitani. Counterexamples of the conjecture on roots of Ehrhart polynomials. Discrete Comput. Geom., 47(3):618-623, 2012. · Zbl 1239.52011 |
| [17] | Alexander M. Kasprzyk. Bounds on fake weighted projective space. Kodai Math. J., 32:197-208, 2009. · Zbl 1216.14047 |
| [18] | Alexander M. Kasprzyk.Classifying terminal weighted projective space. arXiv:1304.3029 [math.AG], 2013. |
| [19] | Alexander M. Kasprzyk and Benjamin Nill. Fano polytopes. In Anton Rebhan, Ludmil Katzarkov, Johanna Knapp, Radoslav Rashkov, and Emanuel Scheidegger, editors, Strings, Gauge Fields, and the Geometry Behind – the Legacy of Maximilian Kreuzer, pages 349-364. World Scientific, 2012. · Zbl 1258.52008 |
| [20] | Maximilian Kreuzer and Harald Skarke. Complete classification of reflexive polyhedra in four dimensions. Adv. Theor. Math. Phys., 4(6):1209-1230, 2000. · Zbl 1017.52007 |
| [21] | I. G. Macdonald. Polynomials associated with finite cell-complexes. J. London Math. Soc. (2), 4:181-192, 1971. · Zbl 0216.45205 |
| [22] | Tetsushi Matsui, Akihiro Higashitani, Yuuki Nagazawa, Hidefumi Ohsugi, and Takayuki Hibi. Roots of Ehrhart polynomials arising from graphs. J. Algebraic Combin., 34(4):721-749, 2011. · Zbl 1229.05122 |
| [23] | Hidefumi Ohsugi and Kazuki Shibata. Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part. Discrete Comput. Geom., 47(3):624-628, 2012. · Zbl 1238.52005 |
| [24] | Miles Reid. Young person’s guide to canonical singularities. In Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), volume 46 of Proc. Sympos. Pure Math., pages 345-414. Amer. Math. Soc., Providence, RI, 1987. · Zbl 0634.14003 |
| [25] | Fernando Rodriguez-Villegas. On the zeros of certain polynomials. Proc. Amer. Math. Soc., 130(8):2251-2254 (electronic), 2002. · Zbl 0992.12001 |
| [26] | Richard P. Stanley. Decompositions of rational convex polytopes. Ann. Discrete Math., 6:333-342, 1980. · Zbl 0812.52012 |
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.
