Explicit rational SOS decompositions for sextic mean forms

Construct explicit rational sum-of-squares decompositions for all sextic mean forms M_{6,p}(X,α) whose SOS membership is established computationally for the specified nonzero partitions and orders p.

Background

For sextic forms, the paper reports SOS membership based on semidefinite-programming decompositions for several nonzero exponent partitions and orders. It explicitly distinguishes these computational verifications from explicit rational certificates, which have not yet been obtained.

References

For p=2,3, SOS membership for all nonzero partitions of~6 is likewise verified computationally; explicit rational decompositions are not yet known.

— The Cone Generated by Positive Semidefinite Mean Polynomials  (2608.22739 - Ghasemi et al., 24 Aug 2026) in Section 4, subsection “Low-degree cases: d=1,2,3,” proof of Theorem M2dp-sos