Truncated normal forms for solving polynomial systems

Abstract

In this poster we present the results of [10]. We consider the problem of finding the common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. We propose a general algebraic framework to find the solutions and to compute the structure of the quotient ring R/I from the cokernel of a resultant map. This leads to what we call Truncated Normal Forms (TNFs). Algorithms for generic dense and sparse systems follow from the classical resultant constructions. In the presented framework, the concept of a border basis is generalized by relaxing the conditions on the set of basis elements. This allows for algorithms to adapt the choice of basis in order to enhance the numerical stability. We present such an algorithm. The numerical experiments show that the methods allow to compute all zeros of challenging systems (high degree, with a large number of solutions) in small dimensions with high accuracy.