Overdetermined Polynomial Homotopies

Abstract:

A polynomial homotopy is a family of polynomial systems with at least one parameter. Continuation methods track solution paths defined by the homotopy. Overdetermined systems, with more equations than unknowns, occur in many applications. For example: we consider systems coming from Schubert problems and systems with positive dimensional singular solution sets. Squaring an overdetermined system with random combinations or with slack variables may introduce spurious solutions. Start solutions for overdetermined homotopies are computed with polyhedral methods, via Puiseux series expansions with tropisms as leading exponents. The Gauss-Newton method is applied in the continuation methods. The algorithms are implemented in PHCpack and phcpy.

The 20th conference of the International Linear Algebra Society, 11-15 July 2016, Leuven, Belgium.

slides of the talk