Strictly positive polynomials in the boundary of the SOS cone (Q6615407)

Consider the sum of square (SOS) cone \(\Sigma_{n,2d}\) of real polynomials in \(n\) variables and of degree \(2d\). In the first four sections of the paper, the authors are interested in strictly positive polynomials on the boundary of this cone and the maximum number of \(\mathbb R\)-linearly independent polynomials of degree \(d\) in the SOS decomposition of such a polynomial.\N\NSections 2 to 4 of the paper is inspired by [\textit{G. Blekherman} et al., SIAM J. Appl. Algebra Geom. 1, No. 1, 175--199 (2017; Zbl 1401.14225)], which provides a lower bound for the Hankel index of the SOS cone for degree \(d\) forms with a specific condition, translating to an upper bound on the number of polynomials in the SOS decomposition of strictly positive polynomials on the boundary. The condition given in that paper is related to the Betti table associated to the image of the \(d\)-th Veronese embedding and can be verified from the Green-Lazarsfeld index. If the Ottaviani-Paoletti (OP) conjecture on the Hankel index holds, then this upper bound is \(\binom{n+d-1}d-(3d-2)\) for \(d\ge 3\) and \(n\ge 3\).\N\NSection 3 of the paper studies the Hilbert function of homogenous ideals \(I\) generated by \(n\) forms of degree \(d\), whose radical is the irrelevant ideal. It asks how many polynomials must be included in \(I\) so that its degree \(k\ge d\) forms are all the forms of degree \(k\). The authors show that the answer can be obtained from the Hilbert polynomial of some of these ideals e.g.\ leading power ideals and ideals containing all monomials in a single variable of degree \(d\). They then show that this result holds for all such ideals if a special case of the Eisenbud-Green-Harris (EGH) conjecture is true. In fact, if EGH is true then one can also obtain an exact value of the Hankel index. However, this section is only complementary and can be entirely skipped without affecting the proofs and main results in the remaining sections.\N\NIn Section 4 the authors prove one of their main results (Proposition 4.17), showing that the upper bound proposed is indeed sharp for \(d\ge 3\) and \(n\ge 3\). Their result depends on a construction inspired by [\textit{G. Blekherman}, J. Am. Math. Soc. 25, No. 3, 617--635 (2012; Zbl 1258.14067)]. The authors also admit that their result still depends on OP being true. It is worth noting that their result is fully proven since their proof would only require OP to hold for the Veronese embedding of the projective plane, which is indeed the case [\textit{C. Birkenhake}, Manuscr. Math. 88, No. 2, 177--184 (1995; Zbl 0857.14030)].\N\NSection 5 addresses the minimum length of the SOS decomposition for strictly positive polynomials on the boundary of the SOS cone. While this is generally challenging, the authors provide several examples where the SOS decomposition is unique up to orthogonal equivalence. Uniqueness can be verified numerically using SEDUMI and Maple or through results from [\textit{G. Blekherman}, J. Am. Math. Soc. 25, No. 3, 617--635 (2012; Zbl 1258.14067)]. A naive dimension count might suggest that strictly positive polynomials in the algebraic boundary of \(\Sigma_{4,6}\) resp. \(\Sigma_{5,4}\) would have minimum SOS decomposition lengths \(5\) resp. \(6\). However, the authors provide examples with unique decomposition of lengths \(6\) resp. \(7\).\N\NAll examples in Section 5 suggest that the minimum SOS decomposition length for strictly positive polynomials in the boundary of the SOS cone is at least the same as the number of variables \(n\). However, in Section 6 the authors conclude with examples of polynomials in the boundary that are strictly positive and have less than \(n\) polynomials in their SOS decompositions.