Minkowski polynomials and mutations

Abstract. Given a Laurent polynomial f, one can form the period of f: this is a func-tion 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 Lau-rent 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 muta-tion 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 re-arrangements 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 man-ifolds, are connected by a sequence of mutations if and only if they have the same pe-riod