Abstract
Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle \(|z+\frac{d}{4}| \le \frac{d}{4}\) or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip \(-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1\). Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ardila, F., Beck, M., Hoşten, S., Pfeifle, J., Seashore, K.: Root polytopes and growth series of root lattices. SIAM J. Discrete Math. 25, 360–378 (2011)
Batyrev, V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994)
Beck, M., De Loera, J.A., Develin, M., Pfeifle, J., Stanley, R.P.: Coefficients and roots of Ehrhart polynomials. Contemp. Math. 374, 15–36 (2005). math/0402148
Bey, C., Henk, M., Wills, J.M.: Notes on the roots of Ehrhart polynomials. Discrete Comput. Geom. 38, 81–98 (2007)
Braun, B.: Norm bounds for Ehrhart polynomial roots. Discrete Comput. Geom. 39, 191–193 (2008)
Braun, B., Develin, M.: Ehrhart polynomial roots and Stanley’s non-negativity theorem. Contemp. Math. 452, 67–78 (2008). math/0610399
CoCoATeam. CoCoA: a system for doing Computations in Commutative Algebra. http://cocoa.dima.unige.it/
Cook, D. II: Nauty in Macaulay2. arXiv:1010.6194 [math]
De Loera, J.A., Haws, D., Hemmecke, R., Huggins, P., Tauzer, J., Yoshida, R.: LattE. http://www.math.ucdavis.edu/~latte/
Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes—Combinatorics and Computation, pp. 43–74. Birkhäuser, Basel (2000)
Harary, F.: Graph Theory. Addison–Wesley, Reading (1969)
Harary, F., Palmer, E.M.: Graphical Enumeration. Academic Press, New York (1973)
Henk, M., Schürmann, A., Wills, J.M.: Ehrhart polynomials and successive minima. Mathematika 52, 1–16 (2005)
Hibi, T.: Algebraic Combinatorics of Convex Polytopes. Carslaw Publications, Glebe (1992)
Hibi, T.: Dual polytopes of rational convex polytopes. Combinatorica 12, 237–240 (1992)
Hibi, T.: A lower bound theorem for Ehrhart polynomials of convex polytopes. Adv. Math. 105, 162–165 (1994)
Köppe, M.: LattE macchiato. http://www.math.ucdavis.edu/~mkoeppe/latte/
Matsui, T.: Development of NZMATH. In: Iglesias, A., Takayama, N. (eds.) Mathematical Software—ICMS 2006. Lecture Notes in Computer Science, vol. 4151, pp. 158–169. Springer, Berlin (2006)
Maxima.sourceforge.net. Maxima, a computer algebra system. http://maxima.sourceforge.net/
NZMATH development group. NZMATH. http://tnt.math.metro-u.ac.jp/nzmath/
Ohsugi, H., Hibi, T.: Normal polytopes arising from finite graphs. J. Algebra 207, 409–426 (1998)
Ohsugi, H., Hibi, T.: Compressed polytopes, initial ideals and complete multipartite graphs. Ill. J. Math. 44(2), 391–406 (2000)
Ohsugi, H., Hibi, T.: Simple polytopes arising from finite graphs. In: Proceedings of the 2008 International Conference on Information Theory and Statistical Learning, pp. 73–79 (2008). arXiv:0804.4287 [math]
Pfeifle, J.: Gale duality bounds for roots of polynomials with nonnegative coefficients. J. Comb. Theory, Ser. A 117(3), 248–271 (2010)
Rodriguez-Villegas, F.: On the zeros of certain polynomials. Proc. Am. Math. Soc. 130, 2251–2254 (2002)
Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)
Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980)
Stanley, R.P.: Enumerative Combinatorics, vol. I. Wadsworth/Cole Advanced Books, Monterey (1986)
Stanley, R.P.: A monotonicity property of h-vectors and h ∗-vectors. Eur. J. Comb. 14, 251–258 (1993)
Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1995). ISBN 0-8218-0487-1
Waterloo Maple Inc. Maple. http://www.maplesoft.com/products/Maple/
Wilson, R.J.: Introduction to Graph Theory, 4th edn. Addison–Wesley, Reading (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Matsui, T., Higashitani, A., Nagazawa, Y. et al. Roots of Ehrhart polynomials arising from graphs. J Algebr Comb 34, 721–749 (2011). https://doi.org/10.1007/s10801-011-0290-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-011-0290-8