Scaling up to multivariate rational function reconstruction

I present an algorithm for the reconstruction of multivariate rational functions from black-box probes. The arguably most important application in high-energy physics is the calculation of multi-loop and multi-leg amplitudes, where rational functions appear as coefficients in the integration-by-parts reduction to basis integrals. I show that for a dense coefficient the algorithm is nearly optimal, in the sense that the number of required probes is close to the number of unknowns.

本文提出了一种从黑盒探针（black-box probes）重构多元有理函数的算法。该算法在高能物理（high-energy physics）领域堪称最具应用价值的场景，便是多圈多腿振幅（multi-loop and multi-leg amplitudes）的计算：在此类计算中，多元有理函数会作为分部积分（integration-by-parts）约化至基积分（basis integrals）过程中的系数形式出现。本文证明，针对稠密系数（dense coefficient）场景，该算法近乎最优，即所需探针的数量与未知量的数目极为接近。