on Newton polytopes, tropisms, and Puiseux series
to solve polynomial systems
Jan Verschelde
Abstract:
Sparse polynomial systems in several variables are characterized
by a set of exponents of monomials that appear with nonzero coefficient.
The convex hull of this set of exponents is the Newton polytope.
To a system of sparse polynomial equations corresponds a tuple of Newton
polytopes.
Vectors perpendicular to a tuple of edges of all polytopes are tropisms
when they define the leading powers of a Puiseux series development for
a solution of the polynomial system. In this talk we outline a polyhedral
approach to solve polynomial systems by means of Puiseux series.
On standard benchmark problems as the cyclic n-roots systems we obtained
exact representations for surfaces of solutions.
This is joint work with Danko Adrovic.
SIAM Conference on Discrete Mathematics.
Minisymposium MS41: Interactions between Computer Algebra and
Discrete Mathematics, 18-21 June 2012, Dalhousie University, Halifax,
Nova Scotia, Canada.
slides of the talk