مرکز منطقه ای اطلاع رساني علوم و فناوري - The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices

Abstract :

Apparently, in any general theory of linear $n$ -dimensional of an systems, it is necessary to exploit the properties of elementary polynomial matrices. In a recent paper [5], it was shown that the internal structure of such matrices could be conceptualized in three distinct but equally meaningful ways. Let $A(z) \\equiv A(z_{1}, z_{2}, \\cdots ,z_{n})$ denote an $m \\times r$ polynomial matrix in the $n$ variables $z_{i}, i = 1 \\rightarrow n$ , where $m \\leq r$ . We say that $A(z)$ is projectively free (PJF), if it can be included as the first $m$ rows of some $r \\times r$ elementary polynomial matrix, that it is unimodular (UM), if there exists an $r \\times m$ polynomial matrix $B(z)$ such that $A(z)B(z) = l_{m}$ , and that it is zero-prime (ZP), if its $^{r}C_{m} m \\times m$ minors are devoid of common zeros. Although it is easily shown that $PJF \\rightarrow UM \\rightarrow ZP$ and that $ZP \\rightarrow UM [5]$ , it was not until recently, in 1976, that the conjecture $UM \\rightarrow PJF$ made by Serre in 1957 was established (independently) by Quillen [7] and Suslin [10]. The two major ideas contained in their proofs are quite remarkable and have a strong synthesis-theoretic flavor. Our purpose in this paper is to explain these ideas by giving an elementary tutorial account of the Quillen-Suslin theorem that is couched completely in the language of polynomials and uses only a minimum of modern abstract algebra. The development presented here applies to polynomials with real or complex coefficients. Those interested in more general coefficient fields (e.g., finite fields) are encouraged to go on to [11]. Throughout we have attempted to present explicit constructions in all proofs. An appendix relates the result $UM \\rightarrow PJF$ to the standard form of the Serre conjecture for projective modules.