A Memory Efficient Variant of an Implementation of the F\(_4\) Algorithm for Computing Gröbner Bases

  • Yun-Ju HuangEmail author
  • Wei-Chih Hong
  • Chen-Mou Cheng
  • Jiun-Ming Chen
  • Bo-Yin Yang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9473)


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 F\(_4\) [9] and F\(_5\) [10] 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.


Gröbner basis F\(_4\) algorithm Time-memory trade-off 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yun-Ju Huang
    • 1
    Email author
  • Wei-Chih Hong
    • 2
  • Chen-Mou Cheng
    • 3
  • Jiun-Ming Chen
    • 4
  • Bo-Yin Yang
    • 5
  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Department of Information Engineering and Computer ScienceFeng Chia UniversityTaichungTaiwan
  3. 3.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan
  4. 4.Department of MathematicsNational Taiwan UniversityTaipeiTaiwan
  5. 5.Institute of Information ScienceAcademia SinicaTaipeiTaiwan

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