On composite abstract homogeneous polynomials

Abstract

We study the null-sets of composite abstract homogeneous polynomials obtained from a pair of abstract homogeneous polynomials defined on a vector space over an algebraically closed field of characteristic zero. First such study for ordinary polynomials in the complex plane was made by Szegö, Cohn, and Egerváry and Szegö’s theorem was later generalized to fields and vector spaces, respectively, by Zervos and Marden. Our main theorem in this paper further generalizes their results and, in the complex plane, improves upon Szegö’s theorem and some other classical results. The method of proof is purely algebraic and utilizes the author’s vector space analogue [Trans. Amer. Math. Soc. 218 (1976), 115-131] of Grace’s theorem on apolar polynomials. We also show that our results cannot be further generalized in certain directions.