COMPUTING A-DISCRIMINANT CHAMBERS AND FASTER HOMOTOPY ALGORITHMS
Bithiah Yuan1, Maurice Rojas2.
University of Hawaii at Hilo, Hilo, HI, 2Texas A&M University, College Station, TX.
The A-discriminant variety is the unique irreducible algebraic hypersurface containing all polynomials (with exponent set contained in A) having singular zero set. While the defining polynomial for the A-discriminant variety is difficult to calculate, it is central in numerous applications, including homotopy algorithms for approximating the real solutions of polynomial systems. We are developing a software package and related quantitative estimates to fully understand the A-discriminant variety for A contained in Zn of cardinality n + 4. Our main goal is to quickly compute which discriminant chamber contains a given polynomial system in order to find homotopies preserving the number of real roots. Understanding the real solutions of polynomial equations will provide applications in numerous disciplines such as robotics and analyzing chemical reactions. (This research was conducted as a part of the 2014 Algorithmic Algebraic Geometry REU at Texas A&M University sponsored by the National Science Foundation.)