Homotopies for positive dimensional solution components of polynomial systems

Abstract:

Motivated by problems from mechanism design, Andrew Sommese and Charles Wampler proposed "numerical algebraic geometry", to deal with positive dimensional solution components of polynomial systems. Recent papers by Sommese, Verschelde, and Wampler developed new homotopies to compute a numerical irreducible decomposition. The positive dimensional solution components are represented by "witness sets", obtained as solutions of the original polynomial equations cut by random linear spaces of complementary dimension. The decomposition into irreducible factors is done by monodromy and certified by linear traces.

Second lecture at RAAG Summer School on Computer Tools for Real Algebraic Geometry, Tuesday 1 July 2003, Rennes, France.