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.
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