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A symmetric pseudo-Boolean function is a map from Boolean tuples to realnumbers which is invariant under input variable interchange. We prove that anysuch function can be equivalently expressed as a power series or factorized.The kernel of a pseudo-Boolean function is the set of all inputs that cause thefunction to vanish identically. Any $n$-variable symmetric pseudo-Booleanfunction $f(x_1, x_2, \dots, x_n)$ has a kernel corresponding to at least one$n$-affine hyperplane, each hyperplane is given by a constraint $\sum_{l=1}^nx_l = \lambda$ for $\lambda\in \mathbb{C}$ constant. We use these results toanalyze symmetric pseudo-Boolean functions appearing in the literature of spinglass energy functions (Ising models), quantum information and tensor networks.

On Symmetric Pseudo-Boolean Functions: Factorization, Kernels and Applications