Some real and unreal enumerative geometry

Workshop in Honor of William Fulton 15 April 2000 Frank Sottile University of Massachusetts-Amherst

   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. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real. Such questions arise also in applications; recently, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real.

   The purpose of this talk is to describe a general method for constructing real solutions to some enumerative problems. This method is based upon homotopy continuation algorithms and is an analog of Viro's method, or toric deformations. We describe how this method may be used to show that some classical problems in enumerative geometry may have all their solutions be real, and apply it to give a classical problem for which we may have no real solutions. Extensive computer experimentation leads to intriguing conjectures on the general situation for many such Schubert-type enumerative problems.