Sum-of-squares certificates for symmetric polynomials on the hypercube: a counterexample to a conjecture of De Klerk and Laurent
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| Format: | Preprint |
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2026
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| _version_ | 1866914621407887360 |
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| author | Polak, Sven |
| author_facet | Polak, Sven |
| contents | 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 $M_{n,n}(x_1-x_1^2,\ldots,x_n-x_n^2)$ of the hypercube is $C_n=1/(n(n+2))$, 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 $C_8\leq 11/1000 <1/80$. In particular, this disproves the conjectured optimal value for $n=8$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_31169 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sum-of-squares certificates for symmetric polynomials on the hypercube: a counterexample to a conjecture of De Klerk and Laurent Polak, Sven Optimization and Control Combinatorics 90C23, 90C22 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 $M_{n,n}(x_1-x_1^2,\ldots,x_n-x_n^2)$ of the hypercube is $C_n=1/(n(n+2))$, 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 $C_8\leq 11/1000 <1/80$. In particular, this disproves the conjectured optimal value for $n=8$. |
| title | Sum-of-squares certificates for symmetric polynomials on the hypercube: a counterexample to a conjecture of De Klerk and Laurent |
| topic | Optimization and Control Combinatorics 90C23, 90C22 |
| url | https://arxiv.org/abs/2605.31169 |