A <polynomial> invariant under changing the sign of each coordinate can have its sum of squares representation averaged over independent <Rademacher random variables> $\varepsilon_i\in\{-1,1\}$. Distinct parity patterns of <monomials> have zero cross terms because $\mathbb E[\prod_i\varepsilon_i^{a_i}]$ vanishes when some exponent is <odd>. For a quadratic <homogeneous polynomial>
$$
q(z)=\sum_i a_i z_i^2+\sum_{i<j}b_{ij}z_iz_j,
$$
this gives
$$
\mathbb E[q(\varepsilon_1z_1,\ldots,\varepsilon_nz_n)^2]
=\left(\sum_i a_i z_i^2\right)^2+\sum_{i<j}b_{ij}^2z_i^2z_j^2.
$$
This identity separates the <even> monomials from each distinct two-coordinate parity pattern.


 Sign averaging of a sum of squares

ID: sign-averaging-of-a-sum-of-squares