(Georgia Institute of Technology, 2023-07-17)
Acevedo Habeych, Jose Gabriel
We compare the cones of symmetric nonnegative polynomials and symmetric sums of squares when the number of variables goes to infinity. We focus on the case of homogeneous polynomials (forms) of fixed even degree in an increasing number of variables. There are different ways of going to infinity but we consider a natural way, given by symmetrization. Symmetrization behaves well with monomial symmetric polynomials, or more precisely their normalizations, or averages, which we call monomial means.
Using methods of the representation theory of the symmetric group we construct, for each n, and in terms of monomial means, a symmetry adapted basis for the space of forms of degree d in n variables. From it we describe both the limit cone of symmetric sums of squares and its dual (the pseudomoment cone). Interestingly, at the limit, monomial means behave like normalized power sums (power means), and these latter give us access to a description of the limit cone dual to symmetric nonnegative forms (the moment cone). Although the latter description is not as explicit as the former, it is enough for allowing us to compare both cones in some of the smallest cases, concluding equality for even symmetric sextics and even symmetric octics, and recovering the equality for symmetric quartics already observed by Blekherman and Riener.
Finally, by computing their logarithmic limit sets (tropicalizations), we show that there are no larger degrees for which both cones coincide. We introduce partial symmetry reduction to give a explicit combinatorial description of the tropical pseudomoment cone, whereas the tropical moment cone can be described via tropical convexity. Furthermore, from the tropicalizations we construct explicit examples of limit nonnegative forms that are not limit sums of squares.