The trace test in numerical algebraic geometry

Numerical algebraic geometry uses tools from numerical analysis
to study algebraic varieties on a computer.  Its origins were
in homotopy methods, which used Newton iterations and homotopy 
continuation to solve systems of equations.   Early homotopy 
algorithm exploited combinatorial structures, such as multihomogeneity,
for efficiency.   In numerical algebraic geometry, a variety X is
represented by a witness sets, which is a linear section of X in
a projective or affine space.

A fundamental step is to decompose a witness set for a variety X
into subsets corresponding to the irreducible components of X
An algorithm for this numerical irreducible decomposition uses 
monodromy to compute a possible decomposition which is verified 
using 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.