Combinatorics of Cremona monomial maps

We study Cremona monomial maps using linear algebra, lattice theory and linear optimization methods. 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. We present another method to compute the inverse of a Cremona monomial map based on integer programming techniques and the notion of a Hilbert basis. 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.