Geometric relations between the zeros of polynomials (Q338509)

The notion of apolarity is extended to complex polynomials of degree at most \(n\) and the corresponding extended Grace theorem is formulated. A new notion of a locus of a polynomial is introduced and its properties are studied. Using this notion, some theorem on the geometry of polynomials can be improved. A refinement of Laguerre's theorem [\textit{Q. I. Rahman} and \textit{G. Schmeisser}, Analytic theory of polynomials. Oxford: Oxford University Press (2002; Zbl 1072.30006)] is also given. Finally six open problems (concerning mainly the loci of polynomials) are posed.