Classification of bounded-degree Polynomial Calculus ideal membership beyond known algebraic cases

Classify bounded-degree Polynomial Calculus ideal membership over finite domains for constraint languages beyond the semilattice, dual-discriminator, and Minority cases, thereby determining the unresolved tractability and lower-bound behavior for degrees d greater than or equal to 1.

Background

The paper completes the Boolean classification but does not provide a corresponding classification over larger domains. It identifies semilattice and dual-discriminator languages as tractable and establishes unconditional lower bounds in the affine/Minority setting, while leaving the general bounded-degree problem unresolved. In particular, decision-problem hardness only yields conditional obstructions for Polynomial Calculus ideal membership, so unconditional lower bounds outside the affine case remain a major missing component.

References

Several cases remain open. Over ${0,1,2}$, the majorities outside the fixed-value family are not classified even for $IMP_d$, and so neither for $PC$-$IMP_d$ (\cref{sect:majorities}); over larger domains, no classification of $PC$-$IMP_d$ is known beyond the semilattice, dual-discriminator, and Minority cases (\cref{sect:csp-pc-imp}).

— Ideal Membership in Polynomial Calculus: Complexity and Reductions  (2609.28243 - Bortolotti et al., 23 Sep 2026) in Section 5, “Future work”

Over ${0,1,2}$, the majorities outside the fixed-value family are not classified even for $IMP_d$, and so neither for $PC$-$IMP_d$ (\cref{sect:majorities}); over larger domains, no classification of $PC$-$IMP_d$ is known beyond the semilattice, dual-discriminator, and Minority cases (\cref{sect:csp-pc-imp}).

— Ideal Membership in Polynomial Calculus: Complexity and Reductions  (2609.28243 - Bortolotti et al., 23 Sep 2026) in Section 5, “Future work”