On birational monomial transformations of plane (Q1774785)

Summary: We study birational monomial transformations of the form \(\varphi(x:y:z)=(\varepsilon_1x^{\alpha_1}y^{\beta_1}z^{\gamma_1}: \varepsilon_2x^{\alpha_2}y^{\beta_2}z^{\gamma_2} : x^{\alpha_3}y^{\beta_3}z^{\gamma_3})\), where \(\varepsilon_1, \varepsilon_2 \in \{-1, 1\}\). These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation \(\varphi\), we prove a formula, which represents the transformation \(\varphi\) as a product of generators of the group. To prove this formula, we use birationally equivalent polynomials \(Ax + By + C\) and \(Ax^p + By^q + Cx^ry^s\). If \(\varphi\) is the transformation which carries one polynomial onto another, then the integral powers of generators in the product, which represents the transformation \(\varphi\), can be calculated by the expansion of \(p/q\) in the continued fraction.