An important problem in mathematics is to find or understand the solutions to a system of multivariate polynomials. The best understanding is obtained when working over the complex numbers, and the situation is more difficult for real solutions. Despite this difficulty, applications demand that we look for real solutions. In this talk, I will report on such work on a specific class of polynomial systems. This is an example of a task we often face---working to bridge the gap between powerful general results and specific assertions needed in applications.
The Schubert calculus of enumerative geometry studies the following polynomial system (expressed geometrically): Given linear subspaces K1 , ... , Kn of an (m+p)-dimensional space with dimKi =m+1-ai and a1 + ... + an = mp, determine the p-planes H with H\cap Ki \neq {0} for each i. Hilbert's 15th problem, now solved, asked for a rigorous justification of Schubert's determination of the number of complex solutions. I will discuss concrete aspects of this; the existence of real solutions, numerical algorithms for finding the solutions, and applications of the understanding gained.