New advances in enumerative real algebraic geometry

Frank Sottile MSRI & University of Wisconsin-Madison Working group in real algebraic geometry Thursday, 12 November 1998 MSRI

In 1984, Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures. In every instance known, the surprising answer is that all may be real. This includes the problem of 3264 conics tangent to 5 plane plane conics, and 2-planes in a vector space satisfying Schubert conditions.

The purpose of this talk is to announce and prove the following breakthrough, namely that for any problem of enumerating p-planes in a vector space having excess intersection with a collection of fixed linear subspaces, there is a choice of the fixed subspaces such that each of the resulting p-planes is real. The fixed planes may be chosen to osculate a rational normal curve at real points. In this way, this result is a case of a remarkable conjecture of Shapiro and Shapiro, and has connections to the pole placement problem in linear systems theory.