Solving systems of equations in supernilpotent algebras

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 $\mu$ 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.

Cited by in corpus (2)

• Chevalley Warning type results on abelian groups
• CC-circuits and the expressive power of nilpotent algebras