Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Characteristic Points of Recursive Systems. | Characteristic Points of Recursive Systems Jason P. Bell Department of Mathematics Simon Fraser University 8888 University Dr. Burnaby BC V5A 1S6 jpb@ Stanley N. Burris Department of Pure Mathematics University of Waterloo Waterloo Ontario N2L 3G1 snburris@ Karen A. Yeats Department of Mathematics Simon Fraser University 8888 University Dr. Burnaby BC V5A 1S6 Submitted May 15 2009 Accepted Aug 18 2010 Published Sep 1 2010 Mathematics Subject Classification 05A16 Abstract Characteristic points have been a primary tool in the study of a generating function defined by a single recursive equation. We investigate the proper way to adapt this tool when working with multi-equation recursive systems. Given an irreducible non-negative power series system with m equations let p be the radius of convergence of the solution power series and let T be the values of the solution series evaluated at p. The main results of the paper include a the set of characteristic points form an antichain in Rm 1 b given a characteristic point a b i the spectral radius of the Jacobian of G at a b is 1 and ii it is 1 iff a b p T c if p T is a characteristic point then i p is the largest a for a b a characteristic point and ii a characteristic point a b with a p is the extreme point p T . 1 Introduction and Preliminaries Recursively defined generating functions play a major role in combinatorial enumeration see the recently published book 9 for numerous examples. The important technique of THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R121 1 expressing a generating function as a product of geometric series as well as other kinds of products was introduced by Euler in the mid 1700s in his study of various problems connected with the number of partitions of integers. This investigation of partition problems was continued by Sylvester and Cayley see for example 5 19 starting in the mid 1850s. The expressions they used for partition generating functions
