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Graphs and Combinatorics

, Volume 35, Issue 1, pp 363–371 | Cite as

Ehrhart Polynomials with Negative Coefficients

  • Takayuki Hibi
  • Akihiro HigashitaniEmail author
  • Akiyoshi Tsuchiya
  • Koutarou Yoshida
Original Paper
  • 34 Downloads

Abstract

It is shown that, for each \(d \ge 4\), there exists an integral convex polytope \({\mathcal {P}}\) of dimension d such that each of the coefficients of \(n, n^{2}, \ldots , n^{d-2}\) of its Ehrhart polynomial \(i({\mathcal {P}},n)\) is negative. Moreover, it is also shown that for each \(d \ge 3\) and \(1 \le k \le d-2\), there exists an integral convex polytope \({\mathcal {P}}\) of dimension d such that the coefficient of \(n^k\) of the Ehrhart polynomial \(i({\mathcal {P}},n)\) of \({\mathcal {P}}\) is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.

Keywords

Integral convex polytope Ehrhart polynomial Positivity problem for combinatorial polynomials 

Mathematics Subject Classification

Primary 52B20 Secondary 52B11 

References

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    Beck, M., Robins, S.: Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Ehrhart, E.: Polynômes Arithmétiques et Méthode des Polyèdres en Combinatoire. Birkhäuser, Boston (1977)zbMATHGoogle Scholar
  3. 3.
    Hibi, T.: Algebraic Combinatorics on Convex Polytopes. Carslaw Publications, Glebe (1992)zbMATHGoogle Scholar
  4. 4.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1. Wadsworth and Brooks, Cole (1986)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  • Takayuki Hibi
    • 1
  • Akihiro Higashitani
    • 2
    Email author
  • Akiyoshi Tsuchiya
    • 1
  • Koutarou Yoshida
    • 1
  1. 1.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan
  2. 2.Department of MathematicsKyoto Sangyo UniversityKyotoJapan

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