Resultants and the Hermite normal form.

摘要

Given two polynomials f and g in Z [x] one can construct the resultant matrix , also called Sylvester's matrix, using the coefficients of f and g. If deg(f) = n and deg(g) = m then the resultant matrix, R, is an (m + n) x (m + n) matrix with the first m rows being the coefficients of f and the last n rows being the coefficients of g, where the leading coefficient of f runs down the main diagonal of the first m rows and the constant term in g continues down the diagonal in the last n rows. This matrix can be used to determine whether f and g share a common polynomial factor of positive degree and if so it can be used to compute their greatest common divisor.; Results about R and its submatrices are discussed in chapter 1. Chapter 2 gives a brief overview of matrix equivalence, and in chapter 3 we see how R relates to the ideal generated by f and g. A polynomial h in this ideal may be represented by a matrix equation in which R is a submatrix of the matrix of coefficients, called the bigradient of f and g. The bigradient is extended to the infinite bigradient by successively adding a row of the coefficients of f and a row of the coefficients of g to obtain an infinite matrix. Results concerning the Hermite normal form of these bigradients and how they relate to the ideal generated by f and g are given.; The fact that Z is a Euclidean domain plays a big role in the results of chapter 3. Hence, in chapter 4 we briefly consider linear combinations of f and g as an ideal in Q [x], as well as polynomials in S[ x] where S is no longer Euclidean.