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Norm bounds for Ehrhart polynomial roots
[article]

2006
*
arXiv
*
pre-print

Stanley found that the

arXiv:math/0602464v2
fatcat:634ocksmlvbqrgcxep7wx5v33u
*roots*of the*Ehrhart**polynomial*of a d-dimensional lattice polytope are*bounded*above in*norm*by 1+(d+1)!. ... We provide an improved*bound*which is quadratic in d and applies to a larger family of*polynomials*. ... In [1] , it was shown that*for*a lattice polytope P of dimension d, the*roots*of L P (t) are*bounded*above in*norm*by 1 + (d + 1)!. ...##
###
Norm Bounds for Ehrhart Polynomial Roots

2008
*
Discrete & Computational Geometry
*

Beck et al. found that the

doi:10.1007/s00454-008-9049-y
fatcat:zys7zz26wrgtpfyco5josnsg6a
*roots*of the*Ehrhart**polynomial*of a d-dimensional lattice polytope are*bounded*above in*norm*by 1 + (d + 1)!. ... In [1] it was shown that*for*a lattice polytope P of dimension d, the*roots*of L P (t) are*bounded*above in*norm*by 1 + (d + 1)!. ... Acknowledgements Thanks to John Shareshian*for*suggestions and advice, Matthias Beck and Sinai Robins*for*introducing me to*Ehrhart*theory, an anonymous referee*for*thoughtful comments, and Laura Braun ...##
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Norm Bounds for Ehrhart Polynomial Roots

2007
*
Discrete & Computational Geometry
*

Beck et al. found that the

doi:10.1007/s00454-006-1297-0
fatcat:llb3h2pkajaedegobohybieutu
*roots*of the*Ehrhart**polynomial*of a d-dimensional lattice polytope are*bounded*above in*norm*by 1 + (d + 1)!. ... In [1] it was shown that*for*a lattice polytope P of dimension d, the*roots*of L P (t) are*bounded*above in*norm*by 1 + (d + 1)!. ... Acknowledgements Thanks to John Shareshian*for*suggestions and advice, Matthias Beck and Sinai Robins*for*introducing me to*Ehrhart*theory, an anonymous referee*for*thoughtful comments, and Laura Braun ...##
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Norm Bounds for Ehrhart Polynomial Roots
[chapter]

*
Twentieth Anniversary Volume:
*

Beck et al. found that the

doi:10.1007/978-0-387-87363-3_11
fatcat:7xj5qrkw6bdsxjjp2rzhoaz72q
*roots*of the*Ehrhart**polynomial*of a d-dimensional lattice polytope are*bounded*above in*norm*by 1 + (d + 1)!. ... In [1] it was shown that*for*a lattice polytope P of dimension d, the*roots*of L P (t) are*bounded*above in*norm*by 1 + (d + 1)!. ... Acknowledgements Thanks to John Shareshian*for*suggestions and advice, Matthias Beck and Sinai Robins*for*introducing me to*Ehrhart*theory, an anonymous referee*for*thoughtful comments, and Laura Braun ...##
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Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem
[article]

2006
*
arXiv
*
pre-print

In this paper, we analyze the

arXiv:math/0610399v1
fatcat:cifnaqq76bhu5hbdhny2xllttq
*root*behavior of general*polynomials*satisfying the conditions of Stanley's theorem and compare this to the known*root*behavior of*Ehrhart**polynomials*. ... Stanley's non-negativity theorem is at the heart of many of the results in*Ehrhart*theory. ...*Norm**Bounds*and Growth Rates In this section we review a*norm**bound*on*roots*of SNN*polynomials*and some results and conjectures about growth rates of*roots*of SNN and*Ehrhart**polynomials*. ...##
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Coefficients and Roots of Ehrhart Polynomials
[article]

2004
*
arXiv
*
pre-print

We prove that

arXiv:math/0402148v1
fatcat:jc6miul7mjhspiwlco2ancnv34
*for*fixed d, there exists a*bounded*region of C containing all*roots*of*Ehrhart**polynomials*of d-polytopes, and that all real*roots*of these*polynomials*lie in [-d, [d/2]). ... We present new linear inequalities satisfied by the coefficients of*Ehrhart**polynomials*and relate them to known inequalities. We also investigate the*roots*of*Ehrhart**polynomials*. ... Ziegler*for*helpful discussions and suggestions. This research was supported in part by the Mathematical Sciences Research Institute. ...##
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Notes on the Roots of Ehrhart Polynomials

2007
*
Discrete & Computational Geometry
*

This improves on the previously best known

doi:10.1007/s00454-007-1330-y
fatcat:cf53quvxtzexdi7o2skg3qdg7a
*bound*n and complements a recent result of Braun where it is shown that the*norm*of a*root*of a*Ehrhart**polynomial*is at most of order n 2 . ... We show that the*Ehrhart**polynomials*of those with one interior lattice point have largest*roots*with*norm*of order n 2 , where n is the dimension. ... Acknowledgements The authors thank the anonymous referee*for*valuable comments and helpful suggestions. ...##
###
Notes on the roots of Ehrhart polynomials
[article]

2006
*
arXiv
*
pre-print

This improves on the previously best known

arXiv:math/0606089v1
fatcat:gcvqu7d2m5gw7agik7ep7i32a4
*bound*n and complements a recent result of Braun where it is shown that the*norm*of a*root*of an*Ehrhart**polynomial*is at most of order n^2. ... We show that the*Ehrhart**polynomials*of those with one interior lattice point have largest*roots*with*norm*of order n^2, where n is the dimension. ...*For*P ∈ P 3 (l), l ≥ 1, the upper*bound*√ 3 on the*norm*of the complex*roots*is only attained by the*roots*of the*Ehrhart**polynomial*of the simplex S 3 (1). The paper is organized as follows. ...##
###
Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

2012
*
Discrete Mathematics & Theoretical Computer Science
*

As a result, we have a smooth Fano polytope that is a counterexample to the two conjectures on the

doi:10.46298/dmtcs.3041
fatcat:xw6lbxwse5hqtmi6jzxugxsrz4
*roots*of*Ehrhart**polynomials*. ... In this extended abstract, we show that if the length of the cycle is 127, then the*Ehrhart**polynomial*has a*root*whose real part is greater than the dimension. ... . • D = 2 ([ • D = In addtition, ] showed that the*norm**bound*of*roots*of the*Ehrhart**polynomial*is O(D 2 ). ...##
###
Counterexamples of the conjecture on roots of Ehrhart polynomials
[article]

2011
*
arXiv
*
pre-print

An outstanding conjecture on

arXiv:1106.4633v2
fatcat:dm4qy26hgfbolfc2goj677r7ii
*roots*of*Ehrhart**polynomials*says that all*roots*α of the*Ehrhart**polynomial*of an integral convex polytope of dimension d satisfy -d ≤(α) ≤ d-1. ... Acknowledgemenets The author would like to thank Hidefumi Ohsugi and Tetsushi Matsui*for*giving him some comments on Example 2.1, pointing out a gap between approximately*roots*and actual*roots*and telling ... Moreover, in [3] , the*norm**bound*of*roots*of the*Ehrhart**polynomial*is given with O(d 2 ). ...##
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Ehrhart Polynomials and Successive Minima

2005
*
Mathematika
*

We investigate the

doi:10.1112/s0025579300000292
fatcat:p3yquhmskbcstafywzr3hhshry
*Ehrhart**polynomial**for*the class of 0-symmetric convex lattice polytopes in Euclidean n-space R^n. ... It turns out that the*roots*of the*Ehrhart**polynomial*and Minkowski's successive minima are closely related by their geometric and arithmetic mean. ... We thank Iskander Aliev, Ulrich Betke and Jesús De Loera*for*helpful discussions. ...##
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Roots of Ehrhart polynomials and symmetric δ-vectors
[article]

2012
*
arXiv
*
pre-print

The conjecture on

arXiv:1112.5777v2
fatcat:a3tgoso7hbdqdfxmos4rcbsya4
*roots*of*Ehrhart**polynomials*, stated by Matsui et al. ... In this paper, we observe the behaviors of*roots*of SSNN*polynomials*which are a wider class of the*polynomials*containing all the*Ehrhart**polynomials*of Gorenstein Fano polytopes. ... Braun [5] gives the best possible*norm**bound*of*roots*of*Ehrhart**polynomials*with O(d 2 ). ...##
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On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series
[article]

2008
*
arXiv
*
pre-print

*For*every positive integer n, consider the linear operator _n on

*polynomials*of degree at most d with integer coefficients defined as follows: if we write h(t)/(1 - t)^d + 1 = ∑_m ≥ 0 g(m) t^m,

*for*some ... Applications are given to

*Ehrhart*δ-

*polynomials*and unimodular triangulations of dilations of lattice polytopes, as well as Hilbert series of Veronese subrings of Cohen--MacCauley graded rings. ... In a similar direction, Brenti-Welker's Theorem 1.1 says that

*for*n sufficiently large, [ 3 , 3 Theorem 1.2(a)], which gives a

*bound*on the

*norm*of the

*roots*of the

*Ehrhart*

*polynomial*of a lattice polytope ...

##
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Fourier transforms of polytopes, solid angle sums, and discrete volume
[article]

2018
*
arXiv
*
pre-print

We also obtain a closed form

arXiv:1602.08593v2
fatcat:neps7xerjrfhzmrr5bzauxoaem
*for*the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter t. ... This closed form generalizes some known results about the Macdonald solid-angle*polynomial*, which is the analogous expression traditionally obtained by requiring that t assumes only integer values. ... Number theorists have applied lattice-point counting inside symmetric bodies in R d to get*bounds*on*norms*of ideals [55] , algebraic geometers have used properties of toric varieties to analyze this ...##
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Roots of Ehrhart polynomials arising from graphs

2011
*
Journal of Algebraic Combinatorics
*

*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 ... 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. ... [3] conjecture that Compared with the

*norm*

*bound*, which is O(D 2 ) in general [5] , the strip in the conjecture puts a tight restriction on the distribution of

*roots*

*for*any

*Ehrhart*

*polynomial*. ...

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