Combinatorics of Cremona monomial maps

TL;DR

This paper explores Cremona monomial maps using combinatorial methods, providing a simple proof that their inverses are also monomial maps of fixed degree and offering explicit formulas for these inverses.

Contribution

It introduces a combinatorial approach to analyze Cremona monomial maps, including an explicit method to determine inverse maps and their degrees, with applications to the plane Cremona monomial group.

Findings

Inverse Cremona monomial maps are also defined by monomials of fixed degree.

Explicit formulas for the monomials defining the inverse maps are provided.

The degree of a Cremona monomial map and its inverse are the same in the plane case.

Abstract

One studies Cremona monomial maps by combinatorial means. Among the results is a simple integer matrix theoretic proof that the inverse of a Cremona monomial map is also defined by monomials of fixed degree, and moreover, the set of monomials defining the inverse can be obtained explicitly in terms of the initial data. A neat consequence is drawn for the plane Cremona monomial group, in particular the known result saying that a plane Cremona (monomial) map and its inverse have the same degree. Included is a discussion about the computational side and/or implementation of the combinatorial invariants stemming from these questions.