Sum-of-squares certificates for symmetric polynomials on the hypercube: a counterexample to a conjecture of De Klerk and Laurent

This paper studies sum-of-squares (SoS) representations of nonnegative polynomials over the hypercube $[0,1{]}^n$. De Klerk and Laurent (SIAM J. Optim., 2010) conjectured that the smallest constant $C_n$ such that the polynomial $x_1\cdots x_n+C_n$ is contained in the degree-$n$ truncated quadratic module $M_{n,n}(x_1-x_1^2,\dots ,x_n-x_n^2)$ of the hypercube is $C_n=1\mathrm{/}(n(n+2))$, for $n$ even. We specialize symmetry reduction techniques for finding sum-of-squares certificates to the hypercube, where the generators $x_i-x_i^2$ are not individually invariant under the symmetric group but form an invariant set, and apply them to this conjecture. Combining this reduction with a further (heuristic) sparsity reduction, a rational rounding step, and an exact verification over $\mathbb{Q}$, we prove the bound $C_8\le 11\mathrm{/}1000<1\mathrm{/}80$. In particular, this disproves the conjectured optimal value for $n=8$.