On polar relations of abstract homogeneous polynomials

Abstract

In this paper we generalize, to vector spaces over algebraically closed fields of characteristic zero, two well-known classical results due to Laguerre and Grace, concerning, respectively, the relative location of the zeros of a complex-valued polynomial and its polar-derivative and the relative location of the zeros of two apolar polynomials. Vector space analogues of their results were generalized, to a certain degree, by Hörmander, Marden, and Zervos. Our results in this paper further generalize their results and, in the complex plane, improve upon those of Laguerre and Grace. Besides, the present treatment unifies their completely independent approaches into an improved and more systematic and abstract theory. We have also shown that our results are best possible in the sense that they cannot be further generalized in certain directions.