Degree-automatability of fixed-degree Sum-of-Squares proofs

Determine whether fixed-degree Sum-of-Squares proofs are degree-automatable over the rationals, meaning that a degree-d proof can be found in time n^{O(d)} whenever one exists, despite the possibility that every such proof requires coefficients of doubly exponential magnitude.

Background

The paper studies Polynomial Calculus ideal-membership certificates partly because a recent Polynomial-Calculus-to-Sum-of-Squares simulation reduces degree-automatability of Sum-of-Squares proofs to solvability of bounded-degree Polynomial Calculus ideal membership. A low-degree Sum-of-Squares proof may exist while all such proofs have coefficients too large for standard semidefinite-programming algorithms to find efficiently. The question is therefore a broader unresolved problem motivating the paper’s tractability results.

References

Whether fixed-degree Sum-of-Squares ($$) proofs are degree-automatable (findable in time $n{O(d)}$ whenever a degree-$d$ proof exists) is a well-known open problem: a system may admit a low-degree $$ proof, yet every such proof can require coefficients of doubly-exponential magnitude in $n$, putting them out of reach of the standard semidefinite programming methods such as the ellipsoid method .

— Ideal Membership in Polynomial Calculus: Complexity and Reductions  (2609.28243 - Bortolotti et al., 23 Sep 2026) in Introduction, subsection “Sum-of-Squares automatability”