A Memory Efficient Variant of an Implementation of the F\(_4\) Algorithm for Computing Gröbner Bases
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 , as well as F\(_4\)  and F\(_5\)  algorithms. The F\(_4\) 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 F\(_4\) algorithm called YAGS (Yet Another Gröbner-basis Solver). YAGS uses less memory than the original algorithm and runs at comparable speed with F\(_4\). Furthermore, YAGS runs even faster than F\(_4\) 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 F\(_4\) over a broad set of parameters and problem sizes.
KeywordsGröbner basis F\(_4\) algorithm Time-memory trade-off
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