Generating A Mathematical Web Service Client With xfy

We have developed a probabilistic method that uses numerical polynomial homotopy continuation to compute Galois groups of Schubert problems. Our implementation of the method produced answers to the questions considered inaccessible by other techniques. The problem of computing all solutions of a system of polynomial equations may be solved using numerical polynomial homotopy continuation [10]. Its basic application finds the solutions of the original system using the solutions to a simpler system with similar structure. At the moment, parallel implementations [6] can crack the systems with about 220 solutions. The goal of this project is a rather unusual application of these methods and software – we intend to use these for pure mathematical research. In this case, the research is on Galois groups of enumerative-geometric problems coming from the Schubert calculus. We present some software to compute Galois groups of many Schubert problems on Grassmannians that take advantage of the parallel implementation of PHCpack [1] and its Maple interface, PHCmaple [9]. We use this software to compute Galois groups of large Schubert problems, in particular, a problem on the Grassmannian of 3-planes in C7 with 91 solutions. These all are found to have Galois group equal to the full symmetric group. The largest Schubert problems whose Galois group that was determined to be the full symmetric group sofar were some with 6 solutions in the Grassmannian of 2 planes in C6 [2].