We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by
\(k\)-element subsets of
\([n]\). We develop a variant of the Gatermann-Parrilo symmetry-reduction method tailored to our setting that allows for several simplifications and a connection to Razborov's flag algebras. We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. This is joint work with James Saunderson, Mohit Singh and Rekha Thomas.
Bienvenue à tous!