Unimodular Valuations beyond Ehrhart

28.11.2023 15:15 - 16:45

Martin Rubey (TU Wien)

Abstract: Consider the set of convex polytopes whose vertices have integer coordinates. A function Z with values in a vector space V is a valuation, if Z(P ∪ Q) = Z(P) + Z(Q) − Z(P ∩ Q) for all polytopes P and Q such that both P ∪ Q and P ∩ Q are also convex polytopes with vertices having integer coordinates. One example for a valuation is the lattice point enumerator, counting the number of lattice points in the polytope, another is the volume. A fundamental result on valuations is the Betke-Kneser theorem. It provides a complete classification of real valued valuations that are invariant with respect to the action of the unimodular group, that is, the general linear group over the integers and translations by integer vectors. In the two dimensional case, the Betke-Kneser theorem states that the area of the polygon, the number of lattice points on the boundary of the polygon and the indicator function which is one if and only if the polygon is non-empty, form a basis of the space of unimodular valuations. We generalize this theorem and classify polynomial valued valuations on lattice polygons that are equivariant with respect to the natural actions of the unimodular group.

I. Fischer, M. Schlosser

BZ 2, 2. OG., OMP 1