An alternative way to establish the necessity part of the classical result on the statistical independence of quadratic forms

This paper provides an alternative to existing proofs of the necessity part of the classical result on the statistical independence of quadratic forms (or of second-degree polynomials) in normal variates. The alternative proof makes use of a lemma on polynomials in a single real variable. This lemma can be regarded as a variation on a result on polynomials in two variables that has been used in the traditional proof. This lemma, which can also be useful in establishing the necessity of the necessary and sufficient condition(s) for a quadratic form (or a second-degree polynomial) to have a noncentral chi-square distribution, is shown to be a consequence of rather elementary properties of polynomials.