Ehrhart theory – bases, transformations, inequalities

In 1962 Ehrhart proved that the number of lattice points in nonnegative integer dilates of a lattice polytope is given by a polynomial in the dilation factor, the Ehrhart polynomial of the lattice polytope.

This result marks the starting point of Ehrhart theory that since then has developed into a very active field of research at the intersection of combinatorics, geometry and algebra.The Ehrhart polynomial encodes fundamental information about the polytope, such as its volume and dimension.

Central questions in Ehrhart theory include the characterization of Ehrhart polynomials and the interpretation of their coefficients. The coefficients in the standard monomial basis can be negative in general. A basis with more desirable properties is the h*-basis which has only nonnegative integer coefficients.

Questions of current particular interest concern the unimodality of the coefficients in the h*-basis as well as Ehrhart polynomials with nonnegative coefficients in the standard monomial basis.

The planned project aims at approaching these and related questions about inequalities satisfied by the coefficients by (i) considering other natural but less studied bases to represent the Ehrhart polynomial, and (ii) gaining a better understanding of the transformations between these bases from a combinatorial and geometric perspective.

Moreover, it is planned to apply and generalize the obtained results to weighted Ehrhart theory and polynomial valuations.