arXiv papers for Max Kölbl

Check out Max's papers on the arXiv. #LatticePolytope #EhrhartTheory #Combinatorics #AlgebraicGeometry
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Minkowski Polynomials and Mutations 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.

"Minkowski Polynomials and Mutations" by Mohammad Akhtar, Tom Coates, Sergey Galkin, and Alexander Kasprzyk. In SIGMA. #AlgebraicGeometry #LatticePolytope #ComputerAlgebra

On the maximum dual volume of a canonical Fano polytope | Forum of Mathematics, Sigma | Cambridge Core We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree (-K_X)^d of a d-dimensional Fano toric variety X with at worst canonical singularities.

"On the maximum dual volume of a canonical Fano polytope" by Gabriele Balletti, Alexander Kasprzyk, and Benjamin Nill. In Forum of Mathematics, Sigma. #AlgebraicGeometry #LatticePolytope

Machine Learning the Dimension of a Polytope We use machine learning to predict the dimension of a lattice polytope directly from its Ehrhart series. This is highly effective, achieving almost 100% accuracy. We also use machine learning to…

"Machine learning the dimension of a polytope" by Tom Coates, Johannes Hofscheier, and Alexander Kasprzyk. #MachineLearning #LatticePolytope #Ehrhart