GASC Seminar "Roots of polynomial systems, Hilbert scheme, and singular points" by Bernard Mourrain (INRIA: Sophia Antipolis, U. Cote d'Azur)
Abstract: A key ingredient in computer algebra for solving polynomial equations is the computation of a normal form, via a good presentation of the ideal generated by these polynomials. This can involve a Groebner basis or border basis computation. From such a normal form, one can then deduce the algebraic structure of the quotient algebra and effectively recover all the solutions of the equations.
We are interested in analyzing normal forms which are known partially. We first present algorithmic characterizations of normal forms based on commuting relations and rank conditions. We show how these lead to efficient methods for finding roots of polynomial equations, which we illustrate by some numerical experimentation.
We show how one can deduce equations for the variety of Artinian algebras associated to r roots with a given basis. We develop the connection with the Hilbert scheme of r points, providing defining equations of degree 2 in the Plucker coordinates of the Grassmannian of r-spaces.
In a more local setting, the characterization of normal forms can be used to define deflated systems of equations defining a multiple root and its inverse system. The singular isolated root of the initial system corresponds to a simple root of this extended system. We show how to exploit it in numerical iterative methods such as Newton iteration to obtain a quadratically converging method to a singular isolated root. Some examples illustrate the approach.
This involves works done in collaboration with Mariemi Alonso, Jerome Brachat, Jonathan Hauenstein, Agnes Szanto, Simon Telen, Marc Van Barel.
Wednesday, October 10, 2018 at 12:15pm to 1:15pm
511 Lake Hall