Degree growth of monomial maps and McMullen's polytope algebra

TL;DR

This paper calculates the dynamical degrees of monomial maps using polytope mixed volumes and links them to McMullen's polytope algebra and toric variety cohomology, revealing invariant classes under certain conditions.

Contribution

It provides a complete computation of dynamical degrees for monomial maps and establishes a novel connection with polytope algebra and toric geometry.

Findings

All dynamical degrees of monomial maps are computed.

Invariant positive cohomology classes are constructed under non-resonance conditions.

The work links polytope algebra with the universal cohomology of toric varieties.

Abstract

We compute all dynamical degrees of monomial maps by interpreting them as mixed volumes of polytopes. By exploiting further the isomorphism between the polytope algebra of P. McMullen and the universal cohomology of complete toric varieties, we construct invariant positive cohomology classes when the dynamical degrees have no resonance.