Quasi-period collapse and 𝐺𝐿_{𝑛}(ℤ)-scissors congruence in rational polytopes

Haase, Christian; McAllister, Tyrrell B.

doi:10.1090/conm/452/08777

Cited by 22 publications

(39 citation statements)

References 6 publications

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“…When P is a Fano polytope, we call r P the Gorenstein index. The minimum period common to the cyclic coefficients of L P * divides r P ; when the period does not equal r P we have a phenomena known as quasi-period collapse [7,16]. In general r P = r Q but, by Proposition 4, P * and Q * are Ehrhart equivalent.…”

Section: Example 8 Consider the Laurent Polynomialsmentioning

confidence: 99%

Minkowski Polynomials and Mutations

Akhtar¹

2012

SIGMA

218

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Abstract. Given a Laurent polynomial f , one can form the period of f : this is a function of one complex variable that plays an important role in mirror symmetry for Fano manifolds. Mutations are a particular class of birational transformations acting on Laurent polynomials in two variables; they preserve the period and are closely connected with cluster algebras. We propose a higher-dimensional analog of mutation acting on Laurent polynomials f in n variables. In particular we give a combinatorial description of mutation acting on the Newton polytope P of f , and use this to establish many basic facts about mutations. Mutations can be understood combinatorially in terms of Minkowski rearrangements of slices of P , or in terms of piecewise-linear transformations acting on the dual polytope P * (much like cluster transformations). Mutations map Fano polytopes to Fano polytopes, preserve the Ehrhart series of the dual polytope, and preserve the period of f . Finally we use our results to show that Minkowski polynomials, which are a family of Laurent polynomials that give mirror partners to many three-dimensional Fano manifolds, are connected by a sequence of mutations if and only if they have the same period.

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Section: Example 8 Consider the Laurent Polynomialsmentioning

confidence: 99%

Minkowski Polynomials and Mutations

Akhtar¹

2012

SIGMA

218

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“…Furthermore, they showed that period collapse never occurs for G dim(P )−1 (P , ·). Haase and McAllister [10] gave a conjectural explanation of period collapse involving splitting the polytope into pieces and applying unimodular transformations onto these pieces.…”

Section: Introductionmentioning

confidence: 99%

Rational Ehrhart quasi-polynomials

Linke

2011

Journal of Combinatorial Theory, Series A

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Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors. It turns out that the number of integral points can still be written as a rational quasi-polynomial. Furthermore, the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.

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“…The article [10] constructs examples for all dimensions ≥ 2 and for arbitrary denominator. For more information, the article [2] constructs simplices whose Ehrhart quasi-polynomial has coefficient functions with prescribed minimum periods, and the article [8] offers some conjectures for why the minimum period of L P (t) is sometimes strictly smaller than the denominator of P.…”

Section: Discussionmentioning

confidence: 99%