Taylor Terms, Constraint Satisfaction and the Complexity of Polynomial Equations over Finite Algebras

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We characterize, within various families of algebras, which of them give rise to an NP-complete problem and which yield a problem solvable in polynomial time. In particular, we prove a dichotomy result which encompasses the cases of lattices, rings, modules, quasigroups and also generalizes a result of Goldmann and Russell for groups [15].