On the complexity of some birational transformations

Using three different approaches, we analyse the complexity of various birational maps constructed from simple operations (inversions) on square matrices of arbitrary size. The first approach comprises the study of the images of lines, and relies mainly on univariate polynomial algebra, the second approach is a singularity analysis and the third method is more numerical, using integer arithmetics. These three methods have their own domain of application, but they give corroborating results, and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions