Counting Square free Cremona monomial maps

TL;DR

This paper employs combinatorial methods to classify and count square-free cubic Cremona maps with up to six variables, extending previous classifications of quadratic Cremona transformations.

Contribution

It introduces a combinatorial approach to classify and enumerate square-free cubic Cremona maps, expanding the understanding of these transformations beyond quadratic cases.

Findings

Reobtained classification of quadratic Cremona transformations

Classified and counted square-free cubic Cremona maps up to six variables

Extended the classification framework to higher-degree maps

Abstract

We use combinatorics tools to reobtain the classification of monomial quadratic Cremona transformations in any number of variables given in \cite{SV2} and to classify and count square free cubic Cremona maps with at most six variables, up to isomorphism.