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Symmetric sums of squares over $$k$$-subset hypercubes. (English) Zbl 1383.05306
Summary: 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 the $$k$$-element subsets of $$[n]$$. For simplicity, we focus on the case $$k=2$$, but our results extend naturally to all values of $$k \geq 2$$. We develop a variant of the Gatermann-Parrilo symmetry-reduction method tailored to our setting that allows for several simplifications and a connection to 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. Our method bypasses much of the technical difficulties needed to apply the Gatermann-Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by A. A. Razborov [J. Symb. Log. 72, No. 4, 1239–1282 (2007; Zbl 1146.03013)] in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting.

##### MSC:
 05D99 Extremal combinatorics 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 20C30 Representations of finite symmetric groups 90C22 Semidefinite programming 90C27 Combinatorial optimization
Flagmatic
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