Numerical algebraic geometry and symbolic computation

Abstract

In a recent joint work with Andrew Sommese and Charles Wampler, numerical homotopy continuation methods have been developed to deal with positive dimensional solution sets of polynomial systems. As solving polynomial systems is such a fundamental problem, connections with recent research in symbolic computation are not hard to find. We will address two such connections.One part of the numerical output of our methods consists of a test used to determine whether a point lies on apositive dimensional solution component. While Grobner bases provide an exact answer to the ideal membership test, geometrical results can be obtained at a lower complexity, as shown by Marc Giusti and Joos Heintz [6]. The recent work of Gregoire Lecerf [7, 9] implements an irreducible decomposition in a symbolic manner.The factorization of multivariate polynomials with approximate coefficients was posed as an open problem in symbolic computation by Erich Kaltofen [8]. Providing a certificate for a numerical factorization by means of the linear trace is related to ideas of Andre Galligo and David Rupprecht [3, 4], which also appears in the works of Tateaki Sasaki [10] and collaborators. See also [1, 2] and [5].