Ehrhart Polynomials with Negative Coefficients

It is shown that, for each d ≥ 4 , there exists an integral convex polytope P of dimension d such that each of the coefficients of n , n 2 , … , n d - 2 of its Ehrhart polynomial i ( P , n ) is negative. Moreover, it is also shown that for each d ≥ 3 and 1 ≤ k ≤ d - 2 , there exists an integral conv...

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Bibliographic Details
Published in:Graphs and combinatorics 2019-01, Vol.35 (1), p.363-371
Main Authors: Hibi, Takayuki, Higashitani, Akihiro, Tsuchiya, Akiyoshi, Yoshida, Koutarou
Format: Article
Language:English
Subjects:
Coefficients
Combinatorics
Engineering Design
Integrals
Mathematics
Mathematics and Statistics
Original Paper
Polynomials
Polytopes
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Summary:It is shown that, for each d ≥ 4 , there exists an integral convex polytope P of dimension d such that each of the coefficients of n , n 2 , … , n d - 2 of its Ehrhart polynomial i ( P , n ) is negative. Moreover, it is also shown that for each d ≥ 3 and 1 ≤ k ≤ d - 2 , there exists an integral convex polytope P of dimension d such that the coefficient of n k of the Ehrhart polynomial i ( P , n ) of 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.
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-018-1990-9