Nonnegative symmetric polynomials and symmetric sums of squares at the limit.

Algebra Student Seminar
Friday, March 4, 2022 - 11:00am for 1 hour (actually 50 minutes)
Skiles 006 and Teams
Jose Acevedo – Georgia Tech –
Jaewoo Jung

Restricting to symmetric homogeneous polynomials of degree 2d we compare the cones of nonnegative polynomials with the cone of sums of squares when the number of variables goes to infinity. We consider two natural notions of limit and for each we completely characterize the degrees for which the limit cones are equal. To distinguish these limit cones we tropicalize their duals, which we compute via tropicalizing spectrahedra and tropical convexity. This gives us convex polyhedral cones which we can completely describe and from them obtain explicit examples of nonnegative symmetric polynomials that are not sums of squares (in some cases for any number >=4 of variables).

This is joint work with Grigoriy Blekherman, Sebastian Debus, and Cordian Riener.


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