Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials. For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman in a recent work. For the cases of more variables or higher degree, the problem is more complicated and very few examples or general results are known. Assuming a conjecture by D. Eisenbud, M. Green, and J. Harris, we obtain bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. In particular, for the case of homogeneous quartic polynomials in 5 variables, for which the required case of the conjecture has been recently proved, we obtain bounds that improve the general bounds known up to date. Finally, combining theoretical results with computational techniques, we find examples and counterexamples that allow us to better understand which of the results obtained by G. Blekherman can be extended to the general case and show that the bounds predicted by our results are attainable.