Numerical irreducible decomposition for multiprojective varieties.

Frank Sottile

A common structure of algebraic equations and varieties from the
applications of mathematics is multihomogeneity, so that the
resulting varieties are subvarieties of products of projective
spaces, called multiprojective varieties.  Hauenstein and
Rodriguez introduced a version of witness sets (witness collections)
for these and an algorithm for their numerical irreducible
decomposition.  This was refined to an optimal algorithm when there
are two factors in work with Leykin and Rodriguez.  is finding a 
version of the trace test that does not suffer from combinatorial 
explosion.

In this talk, I will describe this background and subsequent work
of we four that uses dimension reductions and merging based on
Bertini's Theorem.  This gives a minimal version of multiprojective
witness collections, matroidal witness sets, as well as an optimal
algorithm for numerical irreducible decomposition of such varieties
whose trace test avoids combinatorial explosion.  This is joint work
with Hauenstein, Leykin, and Rodriguez.