A memory efficient variant of an implementation of the F4 algorithm for computing Gröbner bases

Abstract

Solving multivariate systems of polynomial equations is an important problem both as a subroutine in many problems and in its own right. Currently, the most efficient solvers are the Gröbner-basis solvers, which include the XL algorithm [6], as well as F4 [9] and F5 [10] algorithms. The F4 is an advanced algorithm for computing Gröbner bases. However, the algorithm has exponential space complexity and does not provide much flexibility in terms of controlling memory usage. This poses a serious challenge when we want to use it to solve instances of sizes of practical interest. In this paper, we address the issue of memory usage by proposing a variant of F4 algorithm called YAGS (Yet Another Gröbner-basis Solver). YAGS uses less memory than the original algorithm and runs at comparable speed with F4. Furthermore, YAGS runs even faster than F4 when solving dense polynomial systems. In other words, the proposed algorithm can reach better time-memory compromise via deliberately designed techniques to control its memory usage and efficiency. We have implemented a prototype of YAGS and conducted an extensive set of experiments with it. The experiment results demonstrate that the proposed modification does achieve lower time-memory products than the original F4 over a broad set of parameters and problem sizes.