Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacings 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 γ-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).2010 Mathematics Subject Classification. 05A15, 52B12 (primary); 13P10, 26C10, 52B15, 52B20 (secondary).

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“…. , P k and q is in general position with respect to all P i then H q P = H q P 1 ⊔ · · · ⊔ H q P k defines a partition [19]. If P 1 , .…”

Section: 3mentioning

confidence: 99%

Arithmetic Aspects of Symmetric Edge Polytopes

2019

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“…We close with an explicit descriptions of the three cones of combinatorially positive, monotone, and nonnegative lattice-invariant valuations for d = 2. While Stanley's approach made use of the strong ties between Ehrhart polynomials and commutative algebra, our main tool are half-open decompositions introduced by Köppe and Verdolaage [21]. We give a general introduction to translation-invariant valuations in Section 2 and we use halfopen decompositions to give a simple proof of McMullen's result (2) in Section 3.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem

Jochemko

Sanyal

2018

J. Eur. Math. Soc.

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We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h * -vectors. We give a surprisingly simple characterization of combinatorially positive valuations that implies Stanley's nonnegativity and monotonicity of h * -vectors and generalizes work of Beck et al. (2010) from solid-angle polynomials to all translation-invariant simple valuations. For general polytopes, this yields a new characterization of the volume as the unique combinatorially positive valuation up to scaling. For lattice polytopes our results extend work of Betke-Kneser (1985) and give a discrete Hadwiger theorem: There is essentially a unique combinatorially-positive basis for the space of lattice-invariant valuations. As byproducts of our investigations, we prove a multivariate Ehrhart-Macdonald reciprocity and we show universality of weight valuations studied in Beck et al. (2010).

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“…In particular, H q (P ) = P for all q ∈ P . The following result by Köppe and Verdoolaege [23] shows that every polytope can be decomposed into half-open polytopes, and is implicitely also contained in works by Stanley and Ehrhart (see [34]).…”

Section: 1mentioning

confidence: 82%

“…We determine a formula for the h r -tensor polynomial of half-open simplices (Theorem 3.2) by using half-open decompositions of polytopes; an important tool which was introduced by Köppe and Verdoolaege [23]. From this formula and the existence of a unimodular triangulation, we deduce an interpretation of all Ehrhart vectors and matrices of lattice polygons.…”

Section: Introductionmentioning

confidence: 99%

Ehrhart tensor polynomials

Berg

Jochemko

Silverstein

2018

Linear Algebra and its Applications

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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 h r -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 h r -tensor polynomials and discuss possible future research directions.

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Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm

Cited by 44 publications

References 27 publications

Arithmetic Aspects of Symmetric Edge Polytopes

Arithmetic Aspects of Symmetric Edge Polytopes

Combinatorial positivity of translation-invariant valuations and a discrete Hadwiger theorem

Ehrhart tensor polynomials

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