Quadratic Easy Coefficients Conjecture

Prove that for every quadratic rotation symmetric Boolean function, the Hamming-weight sequence admits the Easy Coefficients Conjecture representation in terms of the roots of the characteristic polynomial of its rules matrix: the coefficient of each repeated nonzero root is −1/2, while the coefficient of the root 2 is 1/2, equivalently yielding the formula stated as equation (6) for all positive integers n.

Background

For a rotation symmetric Boolean function, the sequence of Hamming weights as the number of variables varies satisfies a linear recurrence determined by the minimal polynomial of an associated rules matrix. The Easy Coefficients Conjecture asserts that the Hamming weights can be expressed using the roots of the characteristic polynomial with exceptionally simple coefficients: all nonprincipal roots have coefficient −1/2, and the root 2 has coefficient 1/2.

The paper proves this conjecture for quadratic rotation symmetric functions under the additional condition that the minimal and characteristic polynomials of the rules matrix coincide, and separately proves it for quadratic monomial rotation symmetric functions. The general quadratic case remains unresolved when the characteristic polynomial has multiple roots; the paper states that this situation is more difficult because the characteristic-polynomial representation is not uniquely determined by the weight recursion.