Solving Systems of Equations in Supernilpotent Algebras

Recently, M.\ Kompatscher proved that for each finite supernilpotent algebra 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 A, and let \[ d := |A|^{\log_2 (\mu) + \log_2 (|A|) + 1}.\] Applying a method that G.\ K{\'a}rolyi and C.\ Szab\'{o} had used to solve equations over finite nilpotent rings, we show that for A, there is $c \in \N$ 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.