Graduate Computational Algebraic Geometry Seminar
Jan Verschelde
UIC
solving polynomial systems with Puiseux series
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.
Thursday May 2, 2013 at 11:00 AM in SEO 1227