Research On The Lin-Bose Problem And Property Of Gr(?)bner Basis

This thesis mainly focuses on Lin-Bose problem. Based on the greatest common factor of reduced minor and the maximum rank minor of matrix, Lin and Bose brought forward a conjecture of n-ary sub-prime decomposition of polynomial matrix in 1999. In fact, it is a problem about the determinant of a matrix decomposition and matrix factorization.In 2001, Lin and Bose presented four broad Serra's conjectures, and proved the equivalence of conjecture 1-4, in which conjecture 3 is the Lin-Bose conjecture. Lin-Bose conjecture: Let F∈Al×mbe a full row rank matrix, d be the g.c.d. of all the l×lminors of F. If the reduced minors of F generate unit ideal A, then there exists a factorization F = GF1such that det G = dand F1 is a MLP matrix.In the same year, Pommaret employed Quillen-Suslin theorem, projective modules and other methods to prove Lin-Bose conjecture. In 2004, Wang also proved Lin-Bose conjecture by using a simpler and more intuitive method. In fact, Lin-Bose conjecture is a given sufficient condition of a determinant of polynomial matrix decomposition. One hopes to find other better or weaker sufficient conditions or necessary and sufficient conditions. Scholars have been researching and studying this problem actively for many years, but no breakthroughs have been achieved. The problem of Matrix F associated with the decomposition of the factor of f and the property of polynomial matrix has become a hot topic today.On the basis of the Lin-Bose conjecture, this thesis studies the MLP factorization of the l×mmatrix. By employing the 1×1minors and 2×2minors of the l×m matrix, it describes the conditions for the MLP factorization of matrix, and gets some results that broaden the ones from the Lin-Bose conjecture.With the development of Gr(o|¨)bner basis theory and algorithm, people obtain lots of theoretical results and it is also more extensive areas of application. By using Gr(o|¨)bner bases theory and algorithm, this thesis studies the problem of matrix determinant decomposition,factorization, etc and provides a new and more concise way to solve this problem. We propose a new algorithm to determine whether MLP matrix decomposition can be realized by use of the software of Singular.