Parallel degree computation for binomial systems

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• 作者：Tianran Chen^a ; ^{chentia1@msu.edu" class="auth_mail" title="E-mail the corresponding author} ; Dhagash Mehta^b ; ^c ; ^{dmehta@nd.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Binomial systems ; Homotopy continuation ; Algebraic geometry ; BKK root-count ; GPU computing ; Supersymmetric gauge theories
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：March-April 2017
• 年：2017
• 卷：79
• 期：part_P3
• 页码：535-558
• 全文大小：683 K

文摘

Solution sets of systems of binomial equations are of great interest in applied mathematics. For both theoretic and applied purposes, the degree of a solution set (its maximum number of isolated intersections with an affine space of complementary dimension) often plays an important role in understanding its geometric structure. 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. As a case study, the algorithm is applied to the master space problem of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0747717116300554&_mathId=si1.gif&_user=111111111&_pii=S0747717116300554&_rdoc=1&_issn=07477171&md5=47facc240c91e3d5a23f737b2c0a95bc" title="Click to view the MathML source">N=1class="mathContainer hidden">class="mathCode">$\mathcal{N}=1$ gauge theories. The GPU-based implementation achieves nearly 30 fold speedup over its CPU-only counterpart enabling the discovery of previously unknown results.