The trace test in numerical algebraic geometry

Numerical algebraic geometry uses tools from numerical analysis
to study algebraic varieties on a computer.  In numerical algebraic 
geometry, a variety X is represented by a witness set, which is 
a linear section of X in a projective or affine space.
A fundamental step is numerical irreducible decomposition that
decomposes a witness set into subsets corresponding to the 
irreducible components of X using monodromy and the trace test.

In this talk I will introduce numerical algebraic geometry and 
witness sets, and describe numerical irreducible decomposition, 
including a new and elementary proof of the trace test.  I will 
then explain versions of witness sets, the trace test, and numerical 
irreducible decomposition for multihomogeneous varieties X
that take advantage of this structure.

This is joint work with Anton Leykin and Jose Rodriguez.