Determining the number of real solutions to a system of polynomial equations is a challenging problem in symbolic and numeric computation [G-VRRT,St] with real world applications. Related questions include when a problem of enumerative geometry can have all real solutions [So97c] and when may a given physical system be controlled by real output feedback [By]. In May 1995, Boris Shapiro and Michael Shapiro communicated to the author (Frank Sottile) a remarkable conjecture connecting these three lines of inquiry. While this conjecture is false in the original generality, (see Section 5.i) there is very compelling evidence in support of a variant. This is described in some detail in the paper Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro [So98].
This document supplements that paper. While you will find here a brief, self-contained description of the conjectures of Shapiro and Shapiro (and their relevance), the primary goal is to give additional background to the paper, some relevant web links, and present the computational evidence in more detail than was possible in the paper. This includes, as much as possible, documented programs in Maple, SINGULAR, and PHC, which may be run to verify some assertions in the paper, and help the interested reader get started on these explorations.
Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro is a companion to a paper of Verschelde [V] which describes the mathematics involved in a spectacular computations he did to verify some instances of this conjecture. This document concentrates upon the polynomial systems and computational aspects of the conjecture.
That paper and these computations directly inspired a breakthrough in this area of real enumerative geometry. After the paper on Shapiro's conjecture [So98] was written, Sottile proved that for any enumerative problem involving linear subspaces satisfying special Schubert conditions, it is possible to choose real conditions so that all solutions are real [So99]. (A special Schubert condition is the condition that a p-plane meets another linear subspace with excess intersection.) It also holds for some problems of enumerating rational normal curves in a Grassmann variety [So00a], and for a large class of enumerative problems involving flag manifolds [So00b]. Even more recently, A. Gabrielov and A. Eremenko gave a proof of the conjecture of Shapiro and Shapiro for Grassmannians of 2-dimensioinal subspaces of a vector space [EG].