In this talk we consider the Ideal Membership Problem (IMP for short), in which we are given real polynomials $f_0, \dots, f_k$ and the question is to decide whether $f_0$ belongs to the ideal generated by $f_1, \dots, f_k$. In the more stringent version the task is also to find a proof of this fact. The IMP underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. In the majority of such applications the IMP involves so called combinatorial ideals that arise from a variety of discrete combinatorial problems. This restriction makes the IMP significantly easier and in some cases allows for an efficient algorithm to solve it.

The focus of this talk is IMPs arising from Constraint Satisfaction Problems (CSP). First, we discuss that many CSP techniques can be translated to IMPs thus allowing us to significantly improve the methods of studying the complexity of the IMP. We also discuss universal algebraic techniques for the IMP that have been so useful in the study of the CSP. This allows us to prove a general necessary condition for the tractability of the IMP, and three sufficient ones. The sufficient conditions include IMPs arising from systems of linear equations over GF($p$), $p$ prime, and also some conditions defined through special kinds of polymorphisms.

joint work with Andrei Bulatov