Challenges of symbolic computation: my favorite open problems
Journal of Symbolic Computation
A Gröbner free alternative for polynomial system solving
Journal of Complexity
Towards factoring bivariate approximate polynomials
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Semi-numerical determination of irreducible branches of a reduced space curve
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Approximate multivariate polynomial factorization based on zero-sum relations
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Irreducible decomposition of curves
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Approximate factorization of multivariate polynomials via differential equations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Complexity of the resolution of parametric systems of polynomial equations and inequations
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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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 "membership 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 André Galligo and David Rupprecht [3, 4], which also appears in the works of Tateaki Sasaki [10] and collaborators. See also [1, 2] and [5].