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Session S24 - Symbolic Computation: Theory, Algorithms and Applications

Tuesday, July 20, 19:00 ~ 19:25 UTC-3

Strictly positive polynomials in the boundary of the SOS cone

Santiago Laplagne

Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In this talk 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. 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 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. However very few concrete examples are known and hence in many cases it is not possible to determine the optimality of the bounds.

We show that these bounds can also be deduced from a conjecture by D. Eisenbud, M. Green and J. Harris. Combining theoretical results and computational techniques, we find new examples and counterexamples that allow us to prove the optimality of the bound in several cases and better understand which result for the cases studied by G. Blekherman can be extended to the general case.

Joint work with Marcelo Valdettaro (Universidad de Buenos Aires).

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