FiniteFieldSolve: Exactly solving large linear systems in high-energy theory

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap.

大型线性系统在高能理论中扮演着至关重要的角色，它们在振幅自举和积分简化过程中均有出现。本文介绍了FiniteFieldSolve，这是一个用于精确求解有理数域上大型线性系统的通用工具包。该求解器直接与Mathematica接口，安装简便，并能无缝替代Mathematica的原生求解器。在测试中，FiniteFieldSolve的速度比Mathematica快约两个数量级，且内存使用量减少了一个数量级。此外，该软件包在FiniteFieldSolve预定的使用场景中，与其他公开求解器相比亦表现出色。正如软件包的名称所暗示，解决方案是通过已知的有限域方法获得的。这些方法在处理不同素数时，会引入大量的模运算（或整数除法）操作。通过针对每个素数自动重新编译自身，FiniteFieldSolve将除法操作转换成更为快速的指令组合，显著提升了性能。将素数编译的技术可以应用于任何有限域求解器，其中节省的时间将取决于求解器的特性。软件包的操作通过一个振幅自举的详细示例进行了说明。