# Solving systems of equations in supernilpotent algebras

## TL;DR

This paper extends polynomial-time algorithms for solving polynomial equations over finite supernilpotent algebras, providing a more efficient method for finding solutions and implications for circuit satisfiability.

## Contribution

It introduces a new approach that reduces the solution search space for systems of equations in supernilpotent algebras, improving upon previous algorithms.

## Key findings

- Solution testing complexity is reduced to polynomial in the number of variables.
- Provides bounds on the number of assignments needed to find solutions.
- Offers insights into circuit satisfiability problems.

## 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 \[ d := |A|^{\log_2 (\mu) + \log_2 (|A|) + 1}.\] 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 $\mathbf{A}$, 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.