Abstract by Daniel Christensen
Macaulay Matrix Reduction for a Numerical Root-finding Algorithm
A common problem in mathematics is finding the common zeros of a system of n equations in n variables. As general systems of equations may be very difficult to solve, a system of polynomial approximations can be more tractable. H. Michael Moeller and Hans J. Stetter showed that solutions to this problem can be extracted from the eigenvalues of certain Moeller-Stetter matrices which define a linear operator on a certain quotient algebra. One of the fundamental problems which our algorithm aims to solve is the construction of a basis for this quotient algebra. Simon Telen and Marc Van Barel showed that this basis can be constructed by reducing the Macaulay matrix representing the system of polynomial approximations. In this talk, I will discuss various methods for reducing the Macaulay matrix and how to construct a basis for the quotient algebra from the reduced Macaulay matrix.