solving polynomial systems with Puiseux series

Jan Verschelde

Abstract:

The input data for a polynomial system solver consists of a mix of approximate (the coefficients) and exact information (the exponents). The sparse structure of a polynomial is encoded in its Newton polytope, the convex hull spanned by those exponents that appear in monomials with nonzero coefficient. Vectors normal to all edges of the Newton polytopes of the polynomials in a system are called pretropisms. If a solution of an initial form systems defined by a pretropism gives leading coefficients of a series expansion, then the pretropism is a tropism as it defines the leading exponents of the Puiseux series. In our work to extend numerical polyhedral homotopy continuation methods to compute Puiseux series expansions for positive dimensional solution sets, we found a tropical interpretation of Backelin's Lemma for the well known cyclic n-roots problem, for dimensions n = m^2, for any natural number m.
This is joint work with Danko Adrovic.

the 2013 Michigan Computational Algebraic Geometry meeting, 3-4 May 2013, Western Michigan University, Kalamazoo

slides of the talk