International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 32, Pages 1671-1677
doi:10.1155/S0161171204306514
On birational monomial transformations of plane
Department of Mathematics and Statistics, Texas Tech University, Lubbock 79409-1042, TX, USA
Received 20 June 2003; Revised 7 October 2003
Copyright © 2004 Anatoly B. Korchagin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.