General witness sets

Frank Sottile

Numerical algebraic geometry has a close relationship to
intersection theory from algebraic geometry.  We deepen this
relationship, explaining how rational or algebraic equivalence
gives a homotopy.  We present a general notion of witness set
for subvarieties of a smooth complete complex algebraic variety
using ideas from intersection theory.  Under appropriate
assumptions, general witness sets enable numerical algorithms
such as sampling and membership.  These assumptions hold for
products of flag manifolds.  We introduce Schubert witness sets,
which provide general witness sets for Grassmannians and flag
manifolds.