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We study the boundary of the cone of real polynomials that can be decomposedas a sum of squares (SOS) of real polynomials. This cone is included in thecone 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 onthe part of the boundary which is not shared, corresponding to strictlypositive polynomials.  For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4variables, this boundary has been completely characterized by G. Blekherman.For the cases of more variables or higher degree, results by G. Blekherman, R.Sinn and M. Velasco and other authors based on general conjectures give boundsfor the maximum number of polynomials that can appear in a SOS decompositionand the maximum rank of the matrices in the Gram spectrahedron. Combiningtheoretical results and computational techniques, we compute examples thatallow us to prove the optimality of the bounds for all degrees and number ofvariables. Additionally, we give examples for the following problems: examplesin the boundary of the cone that are the sum of less than $n$ squares and havecommon complex roots, and examples of polynomials in the boundary with lengthlarger than the expected from the dimension.

Strictly positive polynomials in the boundary of the SOS cone