Minkowski Polynomials and Mutations

TL;DR

This paper introduces a higher-dimensional generalization of mutations acting on Laurent polynomials, providing a combinatorial framework that preserves key invariants and connects Minkowski polynomials related to Fano manifolds.

Contribution

It develops a combinatorial description of mutations in higher dimensions, linking Minkowski polynomials and Fano polytopes through mutation sequences.

Findings

Mutations preserve the period of Laurent polynomials.

Mutations map Fano polytopes to Fano polytopes.

Minkowski polynomials are connected by mutations if they share the same period.

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.