Parallel degree computation for solution space of binomial systems with an application to the master space of $\mathcal{N}=1$ gauge theories

Abstract

The problem of solving a system of polynomial equations is one of the most fundamental problems in applied mathematics. Among them, the problem of solving a system of binomial equations form a important subclass for which specialized techniques exist. For both theoretic and applied purposes, the degree of the solution set of a system of binomial equations often plays an important role in understanding the geometric structure of the solution set. Its computation, however, is computationally intensive. This paper proposes a specialized parallel algorithm for computing the degree on GPUs that takes advantage of the massively parallel nature of GPU devices. The preliminary implementation shows remarkable efficiency and scalability when compared to the closest CPU-based counterpart. Applied to the "master space problem of $\mathcal{N}=1$ gauge theories" the GPU-based implementation achieves nearly 30 fold speedup over its CPU-only counterpart enabling the discovery of previously unknown results. Equally important to note is the far superior scalability: with merely 3 GPU devices on a single workstation, the GPU-based implementation shows better performance, on certain problems, than a small cluster totaling 100 CPU cores.