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.

翻译：本文研究多项式理想成员问题，该问题针对编码组合问题的理想，这些组合问题是有限语言约束满足问题的实例。在本文中，输入多项式$f$的次数至多为$d=O(1)$（我们称此问题为IMP$_d$）。我们通过证明约束在少数多态下封闭的布尔组合理想的IMP$_d$可在多项式时间内求解，从而弥补了\cite{MonaldoMastrolilli2019}中的研究空白。这完成了对布尔IMP$_d$可解性的完整刻画。我们还证明，对于任意有限域上受双判别器多态约束的问题，其IMP$_d$的成员性证明可在多项式时间内找到。我们的结果可应用于零化子定理和平方和证明等领域。