Author:

Keywords:

Systems of polynomial equations, Multiplication maps, Normal forms,, Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, Polynomial systems, Macaulay matrix, Numerical linear algebra, Multiplication matrices, Normal forms, DECOMPOSITION, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, 0906 Electrical and Electronic Engineering, Numerical & Computational Mathematics, 4613 Theory of computation, 4901 Applied mathematics, 4903 Numerical and computational mathematics

Abstract:

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume that the polynomials are generic in the sense that the number of solutions in $\mathbb{C}^n$ equals the B\'ezout number. The main contribution of this paper is an automated choice of basis for $\mathbb{C}[x]/I$, which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and it depends on the given generators of $I$.