| 1. |
- Bajo, Esme, et al.
(författare)
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Weighted Ehrhart theory: Extending Stanley's nonnegativity theorem
- 2024
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 444
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Tidskriftsartikel (refereegranskat)abstract
- We generalize R. P. Stanley's celebrated theorem that the h⁎-polynomial of the Ehrhart series of a rational polytope has nonnegative coefficients and is monotone under containment of polytopes. We show that these results continue to hold for weighted Ehrhart series where lattice points are counted with polynomial weights, as long as the weights are homogeneous polynomials decomposable as sums of products of linear forms that are nonnegative on the polytope. We also show nonnegativity of the h⁎-polynomial as a real-valued function for a larger family of weights. We explore the case when the weight function is the square of a single (arbitrary) linear form. We show stronger results for two-dimensional convex lattice polygons and give concrete examples showing tightness of the hypotheses. As an application, we construct a counterexample to a conjecture by Berg, Jochemko, and Silverstein on Ehrhart tensor polynomials.
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| 2. |
- Beck, Matthias, et al.
(författare)
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h*-POLYNOMIALS OF ZONOTOPES
- 2019
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Ingår i: Transactions of the American Mathematical Society. - : AMER MATHEMATICAL SOC. - 0002-9947 .- 1088-6850. ; 371:3, s. 2021-2042
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Tidskriftsartikel (refereegranskat)abstract
- The Ehrhart polynomial of a lattice polytope P encodes information about the number of integer lattice points in positive integral dilates of P. The h*-polynomial of P is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the h*-polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the h*-polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients. Furthermore, we present a closed formula for the h*-polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley (1991) on the Ehrhart polynomial. Our results hold not only for h*-polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all h*-polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials.
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| 3. |
- Beck, Matthias, et al.
(författare)
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Lattice zonotopes of degree 2
- 2022
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Ingår i: Beitraege zur Algebra und Geometrie. - : Springer Nature. - 0138-4821 .- 2191-0383.
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Tidskriftsartikel (refereegranskat)abstract
- The Ehrhart polynomialehr P(n) of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers n≥ 0. The degree of P is defined as the degree of its h∗-polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.
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| 4. |
- Beck, Matthias, et al.
(författare)
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Smooth Centrally Symmetric Polytopes in Dimension 3 are IDP
- 2019
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Ingår i: Annals of Combinatorics. - : Springer Publishing Company. - 0218-0006 .- 0219-3094. ; 23:2, s. 255-262
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Tidskriftsartikel (refereegranskat)abstract
- In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda's conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices.
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| 5. |
- Berg, S., et al.
(författare)
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Ehrhart tensor polynomials
- 2018
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 539, s. 72-93
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Tidskriftsartikel (refereegranskat)abstract
- The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix cases, we give Pick-type formulas in terms of triangulations of a lattice polygon. As our main tool, we introduce hr-tensor polynomials, extending the notion of the Ehrhart h⁎-polynomial, and, for matrices, investigate their coefficients for positive semidefiniteness. In contrast to the usual h⁎-polynomial, the coefficients are in general not monotone with respect to inclusion. Nevertheless, we are able to prove positive semidefiniteness in dimension two. Based on computational results, we conjecture positive semidefiniteness of the coefficients in higher dimensions. Furthermore, we generalize Hibi's palindromic theorem for reflexive polytopes to hr-tensor polynomials and discuss possible future research directions.
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| 6. |
- Bränden, Petter, 1976-, et al.
(författare)
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The Eulerian Transformation
- 2022
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Ingår i: Transactions of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9947 .- 1088-6850. ; 375:3, s. 1917-1931
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Tidskriftsartikel (refereegranskat)abstract
- Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation A : R[t] -> R[t] defined by A(t(n)) = A(n)(t), where A(n)(t) denotes the n-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator A, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.
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| 7. |
- Dostert, Maria, et al.
(författare)
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Learning Polytopes with Fixed Facet Directions
- 2023
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Ingår i: SIAM Journal on Applied Algebra and Geometry. - : Society for Industrial & Applied Mathematics (SIAM). - 2470-6566. ; 7:2, s. 440-469
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Tidskriftsartikel (refereegranskat)abstract
- We consider the task of reconstructing polytopes with fixed facet directions from finitely many support function evaluations. We show that for a fixed simplicial normal fan, the least-squares estimate is given by a convex quadratic program. We study the geometry of the solution set and give a combinatorial characterization for the uniqueness of the reconstruction in this case. We provide an algorithm that, under mild assumptions, converges to the unknown input shape as the number of noisy support function evaluations increases. We also discuss limitations of our results if the restriction on the normal fan is removed.
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| 8. |
- Ferroni, Luis, et al.
(författare)
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Ehrhart polynomials of rank two matroids
- 2022
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Ingår i: Advances in Applied Mathematics. - : Elsevier BV. - 0196-8858 .- 1090-2074. ; 141, s. 102410-
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Tidskriftsartikel (refereegranskat)abstract
- Over a decade ago De Loera, Haws and Koppe conjectured that Ehrhart polynomials of matroid polytopes have only positive coefficients and that the coefficients of thecorresponding h*-polynomials form a unimodal sequence. The first of these intensively studied conjectures has recently been disproved by the first author who gave counterexamples in all ranks greater than or equal to three. In this article we complete the picture by showing that Ehrhart polynomials of matroids of lower rank have indeed only positive coefficients. Moreover, we show that they are coefficient-wise bounded by the Ehrhart polynomials of minimal and uniform matroids. We furthermore address the second conjecture by proving that h*-polynomials of matroid polytopes of sparse paving matroids of rank two are real-rooted and therefore have logconcave and unimodal coefficients. In particular, this shows that the h*-polynomial of the second hypersimplex is realrooted, thereby strengthening a result of De Loera, Haws and Koppe.
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| 9. |
- Higashitani, Akihiro, et al.
(författare)
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ARITHMETIC ASPECTS OF SYMMETRIC EDGE POLYTOPES
- 2019
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Ingår i: Mathematika. - 0025-5793 .- 2041-7942. ; 65:3, s. 763-784
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Tidskriftsartikel (refereegranskat)abstract
- We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Grobner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the h*-polynomial in case of complete bipartite graphs. In particular, we show that the h*-polynomial is gamma-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthening due to Nevo and Petersen [On gamma-vectors satisfying the Kruskal-Katona inequalities. Discrete Comput. Geom. 45(3) (2011), 503 521].
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| 10. |
- Jochemko, Katharina, et al.
(författare)
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Combinatorial mixed valuations
- 2017
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Ingår i: Advances in Mathematics. - : Academic Press. - 0001-8708 .- 1090-2082. ; 319, s. 630-652
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Tidskriftsartikel (refereegranskat)
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