Spectral Realizability of Nonnegative Companion Matrices via Saddle-Point Asymptotics

Given a finite set S = {λ1, . . . , λn} ⊂ C of distinct complex numbers, when do there exist positive integer multiplicities m1, . . . , mn such that P(x) = ∏ᵢ₌₁ⁿ (x − λᵢ)^mᵢ has every coefficient nonnegative? Equivalently, when can S be realized, after raising algebraic multiplicities, as the spectrum of a nonnegative companion matrix? We give a complete answer: such multiplicities exist if and only if (1) S is closed under complex conjugation, (2) S contains no strictly positive real number, and (3) whenever S contains a point of positive real part it also contains one of negative real part. The necessity direction is elementary. The sufficiency direction is the substantial part of the paper and combines two independent quantitative tools, each developed here with a complete self-contained proof: a uniform local limit theorem controlling the hypergeometric distribution's convergence to the binomial distribution with an explicit O(1/N) error uniform in the coefficient index, and an ultimate-positivity theorem, proved via Olver's parameter-uniform saddle-point method, for the coefficients of f(x)(x² + bx + c)^m as m → ∞. We assemble these into a full proof of sufficiency, correcting two boundary-case and exponent-bookkeeping gaps present in earlier drafts of this argument. The resulting characterization sits naturally in nonnegative matrix theory, since a nonnegative companion matrix realizing P(x) is, up to a change of basis, a nonnegative matrix with characteristic polynomial P.