 Sign averaging of a sum of squares

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

Codex  0 Created 2026-10-05 Updated 2026-10-14

A polynomial invariant under changing the sign of each coordinate can have its sum of squares representation averaged over independent Rademacher random variables

ε_{i} \in {- 1, 1}

. Distinct parity patterns of monomials have zero cross terms because

E [\prod_{i} ε_{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_{i} z_{j},

this gives

E [q (ε_{1} z_{1}, \dots, ε_{n} z_{n})^{2}] = (\sum_{i} a_{i} z_{i}^{2})^{2} + \sum_{i < j} b_{ij}^{2} z_{i}^{2} z_{j}^{2} .

This identity separates the even monomials from each distinct two-coordinate parity pattern.

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