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Norm bounds for Ehrhart polynomial roots [article]

Benjamin Braun
2006 arXiv   pre-print
Stanley found that the 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)!.  ... 
arXiv:math/0602464v2 fatcat:634ocksmlvbqrgcxep7wx5v33u

Norm Bounds for Ehrhart Polynomial Roots

Benjamin Braun
2008 Discrete & Computational Geometry  
Beck et al. found that the 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  ... 
doi:10.1007/s00454-008-9049-y fatcat:zys7zz26wrgtpfyco5josnsg6a

Norm Bounds for Ehrhart Polynomial Roots

Benjamin Braun
2007 Discrete & Computational Geometry  
Beck et al. found that the 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  ... 
doi:10.1007/s00454-006-1297-0 fatcat:llb3h2pkajaedegobohybieutu

Norm Bounds for Ehrhart Polynomial Roots [chapter]

Benjamin Braun
Twentieth Anniversary Volume:  
Beck et al. found that the 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  ... 
doi:10.1007/978-0-387-87363-3_11 fatcat:7xj5qrkw6bdsxjjp2rzhoaz72q

Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem [article]

Benjamin Braun, Mike Develin
2006 arXiv   pre-print
In this paper, we analyze the 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.  ... 
arXiv:math/0610399v1 fatcat:cifnaqq76bhu5hbdhny2xllttq

Coefficients and Roots of Ehrhart Polynomials [article]

M. Beck, J. A. De Loera, M. Develin, J. Pfeifle, R. P. Stanley
2004 arXiv   pre-print
We prove that 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.  ... 
arXiv:math/0402148v1 fatcat:jc6miul7mjhspiwlco2ancnv34

Notes on the Roots of Ehrhart Polynomials

Christian Bey, Martin Henk, Jorg M. Wills
2007 Discrete & Computational Geometry  
This improves on the previously best known 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.  ... 
doi:10.1007/s00454-007-1330-y fatcat:cf53quvxtzexdi7o2skg3qdg7a

Notes on the roots of Ehrhart polynomials [article]

Christian Bey, Martin Henk, Joerg M. Wills
2006 arXiv   pre-print
This improves on the previously best known 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.  ... 
arXiv:math/0606089v1 fatcat:gcvqu7d2m5gw7agik7ep7i32a4

Smooth Fano polytopes whose Ehrhart polynomial has a root with large real part

Hidefumi Ohsugi, Kazuki Shibata
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 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 ).  ... 
doi:10.46298/dmtcs.3041 fatcat:xw6lbxwse5hqtmi6jzxugxsrz4

Counterexamples of the conjecture on roots of Ehrhart polynomials [article]

Akihiro Higashitani
2011 arXiv   pre-print
An outstanding conjecture on 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 ).  ... 
arXiv:1106.4633v2 fatcat:dm4qy26hgfbolfc2goj677r7ii

Ehrhart Polynomials and Successive Minima

Martin Henk, Achill Schürmann, Jörg M. Wills
2005 Mathematika  
We investigate the 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.  ... 
doi:10.1112/s0025579300000292 fatcat:p3yquhmskbcstafywzr3hhshry

Roots of Ehrhart polynomials and symmetric δ-vectors [article]

Akihiro Higashitani
2012 arXiv   pre-print
The conjecture on 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 ).  ... 
arXiv:1112.5777v2 fatcat:a3tgoso7hbdqdfxmos4rcbsya4

On the Log-Concavity of Hilbert Series of Veronese Subrings and Ehrhart Series [article]

Matthias Beck, Alan Stapledon
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  ... 
arXiv:0804.3639v3 fatcat:5b2zazhvnncu7fc6yke5nnyv5y

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

Ricardo Diaz, Quang-Nhat Le, Sinai Robins
2018 arXiv   pre-print
We also obtain a closed form 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  ... 
arXiv:1602.08593v2 fatcat:neps7xerjrfhzmrr5bzauxoaem

Roots of Ehrhart polynomials arising from graphs

Tetsushi Matsui, Akihiro Higashitani, Yuuki Nagazawa, Hidefumi Ohsugi, Takayuki Hibi
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.  ... 
doi:10.1007/s10801-011-0290-8 fatcat:razmiui265ehhatm2nos4wglrq
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