**Probabilistic analysis on Macaulay matrices over finite fields and complexity of constructing Gröbner bases**

*Igor Semaev and Andrea Tenti*

**Abstract: **Gröbner basis methods are used to solve systems of polynomial equations over finite fields, but their complexity is poorly understood. In this work an upper bound on the time complexity of constructing a Gröbner basis and finding a solutions of a system is proved. A key parameter in this estimate is the degree of regularity of the leading forms of the polynomials. Therefore, we provide an upper bound on the degree of regularity for a sufficiently overdetermined system of forms over any finite field. The bound holds with probability tending to 1 and depends only on the number of variables, the number of polynomials, and their degrees. Our results imply that sufficiently overdetermined systems of polynomial equations are solvable in polynomial time with high probability.

**Category / Keywords: **foundations / Polynomial equation systems, finite fields, Macaulay matrices, Groebner bases, multisets, complexity

**Original Publication**** (in the same form): **Journal of Algebra
**DOI: **10.1016/j.jalgebra.2020.08.035

**Date: **received 5 Aug 2019, last revised 2 Oct 2020

**Contact author: **andrea tenti at uib no

**Available format(s): **PDF | BibTeX Citation

**Version: **20201002:124449 (All versions of this report)

**Short URL: **ia.cr/2019/903

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