Abstract
We consider the polynomial Ideal Membership Problem (IMP) for ideals encoding combinatorial problems that are instances of CSPs over a finite language. In this paper, the input polynomial $f$ has degree at most $d=O(1)$ (we call this problem IMP$_d$). We bridge the gap in \cite{MonaldoMastrolilli2019} by proving that the IMP$_d$ for Boolean combinatorial ideals whose constraints are closed under the minority polymorphism can be solved in polynomial time. This completes the identification of the tractability for the Boolean IMP$_d$. We also prove that the proof of membership for the IMP$_d$ for problems constrained by the dual discriminator polymorphism over any finite domain can be found in polynomial time. Our results can be used in applications such as Nullstellensatz and Sum-of-Squares proofs.