Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: P´lya’s Permanent Problem o. | Polya s Permanent Problem William McCuaig 5268 Eglinton St. Burnaby British Columbia Canada V5G 2B2 Submitted Dec 5 1998 Accepted Feb 17 2004 Published Nov 6 2004 MR Subject Classifications Primary 05C20 Secondary 05-02 05A15 05C15 05C38 05C70 05C75 05C83 15A15 Abstract A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero negative and positive entries. We give structural characterizations of sign-nonsingular matrices digraphs with no even length dicycles and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations which are topological in nature imply polynomial algorithms. 1 Introduction Polya s permanent problem can be traced back to an innocent exercise from 1913. 1 There are many equivalent versions such as characterizing when det B perm B for a square nonnegative real matrix B when all dicycles of a digraph have odd length or when a square real matrix is sign-nonsingular. In this section we start with some basic definitions concepts and theorems. Then we briefly illustrate why the versions of the previous paragraph are equivalent. Following this we state the Main Theorem which solves all versions of Polya s permanent problem. Finally we outline the contents of the rest of the paper. There are some nonstandard figure conventions used in this paper. The notation F. N appears in the text when Figure number N is relevant. There are variations such as F. Ni p. x where part i of Figure N on page x is relevant. If part i of Figure N is a graph we refer to it as HNi. Further figure conventions are explained in the second paragraphs of Sections 6 and 8. We assume the reader is familiar with elementary linear algebra and complexity theory. For an informal discussion of complexity theory see Lovasz and Plummer 28 or Plummer 35 . We will use the graph terminology of Bondy and Murty 3 . Support from NSERC is gratefully acknowledged. 1 Quoted from Brualdi and Shader 6 . THE .
