Strong membership complexity for the SOS cone

Determine the time complexity of the strong membership problem for the cone of SOS polynomials of degree at most an even number d≥2.

Background

The paper distinguishes weak and strong membership problems for the cone of sum-of-squares polynomials. It cites a result establishing polynomial-time solvability of weak membership, while explicitly stating that the exact complexity of strong membership remains unresolved. Resolving this question would clarify the computational status of exact or robust certification of SOS-polynomial membership.

References

G\"artner, Magron, and Vallentin recently showed that the weak membership problem for the cone of SOS polynomials of degree at most an even number $d\geq2$ can be solved in polynomial time. To the best of our knowledge, the time complexity of the strong membership problem is still unknown.

— Further analysis and extension of the higher-order Newton method of Ahmadi, Chaudhry, and Zhang  (2609.01001 - Voert et al., 1 Sep 2026) in Section 2, subsection “SOS-convex polynomial”