On Reznick's Question about the Cone of Forms with a Sum-of-Squares Odd Power: Explicit Non-Convexity for Sextics in Five and Quartics in Seven Variables

For an even degree \(m\) and an odd exponent \(2k+1\), let \(\Sigma_{n,m}(2k+1)\) be the set of real forms \(f\) of degree \(m\) in \(n\) variables such that \(f^{2k+1}\) is a sum of squares of forms. B. Reznick asked (Oberwolfach Report 14/2023, p. 779) whether this set is a closed convex cone. It is a closed cone; whether it is convex is still stated as open by Blekherman, Kozhasov and Reznick (Forum Math. Sigma 14 (2026), e65). A recent preprint of A. Kriebel (Zenodo, 6 October 2026) answers the question in the negative: it proves that \(\Sigma_{n,6}(3)\) is not convex for \(n=3\cdot10^{62}\), and that for every odd exponent \(\ge3\) the set of sextics is not convex in some dimension, by averaging many copies of one seed form written in separate variables. The first negative answer is therefore due to that preprint, and not to this note. The examples of the present note were found after that preprint had been read, and they use its principle with two copies instead of \(10^{62}\). We give explicit forms \(p\), \(q\) such that \(p^3\) and \(q^3\) are sums of squares and \((p+q)^3\) is not: the sextics \(p=M_c(x,y,z)\), \(q=M_c(u,v,z)\) in five variables, with \(M_c=x^4y^2+x^2y^4+z^6-c\,x^2y^2z^2\) and \(c=\frac{41}{16}\), and the quartics \(p=Q_c(x,y,z,w)\), \(q=Q_c(x',y',z',w)\) in seven variables, with \(Q_c=w^4+x^2y^2+y^2z^2+z^2x^2-c\,xyzw\) and \(c=\frac{13}4\). Hence \(\Sigma_{n,m}(3)\) is not convex for \(n\ge5\) and even \(m\ge6\), and for \(n\ge7\) and even \(m\ge4\); and, by the many-copies argument of Kriebel's preprint, for every even \(m\ge4\) and every odd exponent \(\ge3\) the set is not convex in all sufficiently large dimensions. The two main theorems are computer-assisted: the proofs rest on exact rational certificates (Gram matrices, and a functional that is non-negative on the relevant squares and negative on \((p+q)^3\)), found by numerical semidefinite programming and verified in exact arithmetic by a short program. As a by-product, \(M_c^3\) is a sum of squares for all \(c\le\frac{41}{16}=2.5625\) (proved before: \(c\le(15/13)^{1/3}\approx1.0489\); reported from experiments: up to \(c\approx2.56548\)). The cases \(n=3,4\) with even \(m\ge6\), the quartics with \(n=4,5,6\), and explicit small dimensions for exponents at least \(5\) remain open; numerical searches found no example there, which is a test and not a proof. This is an unrefereed note.