tailieunhanh - Báo cáo toán học: " Reduced Canonical Forms of Stoppers"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Reduced Canonical Forms of Stoppers. | Reduced Canonical Forms of Stoppers Aaron N. Siegel Mathematical Sciences Research Institute 17 Gauss Way Berkeley CA 94720 Submitted May 12 2006 Accepted Jun 26 2006 Published Jul 28 2006 Mathematics Subject Classification 91A46 Abstract The reduced canonical form of a loopfree game G is the simplest game infinitesimally close to G. Reduced canonical forms were introduced by Calistrate and Grossman and Siegel provided an alternate proof of their existence. In this paper we show that the Grossman-Siegel construction generalizes to find reduced canonical forms of certain loopy games. 1 Introduction The reduced canonical form G of a loopfree game G is the simplest game infinitesimally close to G. Reduced canonical forms were introduced by Calistrate 4 who suggested a construction for G. Calistrate s construction was recently proved correct by Grossman and Siegel 6 who also gave a second quite different construction of G. In this paper we show that the Grossman-Siegel construction generalizes to a class of loopy games known as stoppers. The arsenal of tools available in the study of loopy games is currently rather limited. The temperature theory and the theory of atomic weights two of the most familiar techniques used to attack loopfree games have not yet been adequately generalized to stoppers though Berlekamp and others have extended the temperature theory to many loopy Go positions 2 . The reduced canonical form is therefore a welcome addition to the theory. It has already proven to be a useful tool in the study of loopfree games particularly in situations where temperatures and atomic weights yield little information see for example 7 Section 7 . In Section 2 we review the basic theory of stoppers and develop the necessary machinery for carrying out the reduction argument. Section 3 presents the construction. Finally Section 4 poses some interesting open problems and directions for further research. THE ELECTRONIC JOURNAL OF COMBINATORICS