International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 29-32, Pages 1671-1677
doi:10.1155/S0161171204306514

On birational monomial transformations of plane

Abstract

We study birational monomial transformations of the form φ(x:y:z)=(ϵ1xα1yβ1zγ1:ϵ2xα2yβ2zγ2:xα3yβ3zγ3), where ϵ1,ϵ2∈{−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation φ, we prove a formula, which represents the transformation φ as a product of generators of the group. To prove this formula, we use birationally equivalent polynomials Ax+By+C and Axp+Byq+Cxrys. If φ is the transformation which carries one polynomial onto another, then the integral powers of generators in the product, which represents the transformation φ, can be calculated by the expansion of p/q in the continued fraction.