In generalizing Grace’s
Theorem on apolar polynomials, it was convenient (see reference 1) to use
a set of relations among the coefficients of a pair of polynomials which is
invariant under nonsingular linear transformations of the polynomials. Other
invariant relations among their coefficients will define other classes of pairs of
polynomials. The present paper establishes a set of relations which is both
necessary and sufficient to guarantee that two polynomials of degree n either
have a common zero of multiplicity at least n − 1 or have their zeros all
lying on one circle and so related that if the zeros of one polynomial are
transformed, by a linear transformation, into the n-th roots of +1 then the zeros of
the other are carried, by the same transformation, into the n-th roots of
−1.
