Explicit rational SOS decomposition for the uniform quartic mean form

Construct an explicit rational sum-of-squares decomposition for the quartic mean form M_{4,1}(X,(1,1,1,1)).

Background

The paper establishes SOS membership for the uniform four-variable quartic form M_{4,1}(X,(1,1,1,1)) by semidefinite-programming verification and notes that the form is nonnegative by convexity. However, the computational certificate is not presented as an explicit rational decomposition, leaving the construction of such a decomposition unresolved.

References

For n=4 with uniform weights α=(1,1,1,1), nonnegativity follows from convexity of t↦t4; SOS membership is verified by SDP decomposition (a rank-3 Gram matrix); an explicit rational decomposition is open.

— 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

Does Theorem~\ref{thm:M2dp-sos} extend to arbitrary d?

— The Cone Generated by Positive Semidefinite Mean Polynomials  (2608.22739 - Ghasemi et al., 24 Aug 2026) in Remark rem:open, Section 4