Vol. 17, No. 1, 1966

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A certain class of polynomials

Ruth Goodman

Vol. 17 (1966), No. 1, 57–69

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.

Mathematical Subject Classification
Primary: 30.10
Received: 8 August 1964
Revised: 8 November 1964
Published: 1 April 1966
Ruth Goodman