  A symmetric pseudo-Boolean function is a map from Boolean tuples to real
numbers which is invariant under input variable interchange. We prove that any
such 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 the
function to vanish identically. Any $n$-variable symmetric pseudo-Boolean
function $f(x_1, x_2, \dots, x_n)$ has a kernel corresponding to at least one
$n$-affine hyperplane, each hyperplane is given by a constraint $\sum_{l=1}^n
x_l = \lambda$ for $\lambda\in \mathbb{C}$ constant. We use these results to
analyze symmetric pseudo-Boolean functions appearing in the literature of spin
glass energy functions (Ising models), quantum information and tensor networks.
