Equations in universal algebra

Wider research context / theoretical framework We want to solve equations and to describe their solutions over arbitrary algebraic structures. This is a topic in universal algebra with links to circuit complexity. Hypotheses/research questions/objectives Main questions are: Solving equations: provide algorithms for solving equations over certain classes algebras such as solvable groups. Polynomial systems and quasi-identities: investigate the complexity of checking the validity of quasi-identities. The geometry of a universal algebra: find lower bounds for the number of solutions, characterize those collections that are the algebraic geometry of some algebra. Notions of solvability: How much must an algebra be extended to contain a solution of a given equation? Approach/methods Solving equations: we want to adapt the “hitting set” strategy which consists in finding for every equation a small set T such that if the equation has a solution, it has a solution in T. To this end, we investigate an algebra’s ability to express logical conjunctions by term functions using a concept of “degree” of a function that we recently developed. The geometry of a universal algebra: we want to generalize Warning’s Second Theorem on the number of solutions to an universal algebraic framework, and we want to find methods for determining logical connections between equations by determining the validity of quasi-identities. For the description of algebraic geometries, we want to use analogies with clone theory. Notions of solvability: we want to start from noncommutative rings with unit and find the universal algebraic properties that distinguish noncommutative rings from finite fields.