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arXiv · 1901.07862

Solving systems of equations in supernilpotent algebras

Erhard Aichinger
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Abstract

Recently, M. Kompatscher proved that for each finite supernilpotent algebra $\mathbf{A}$ in a congruence modular variety, there is a polynomial time algorithm to solve polynomial equations over this algebra. Let $μ$ be the maximal arity of the fundamental operations of $\mathbf{A}$, and let \[ d := |A|^{\log_2 (μ) + \log_2 (|A|) + 1}.\] Applying a method that G. Károlyi and C. Szabó had used to solve equations over finite nilpotent rings, we show that for $\mathbf{A}$, there is $c \in \mathbb{N}$ such that a solution of every system of $s$ equations in $n$ variables can be found by testing at most $c n^{sd}$ (instead of all $|A|^n$ possible) assignments to the variables. This also yields new information on some circuit satisfiability problems.

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BibTeXRIS

Erhard Aichinger. 2019-01-23. Solving systems of equations in supernilpotent algebras. https://doi.org/10.4230/lipics.mfcs.2019.72

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