zeros and critical points of polynomials apolarity polar derivative Grace theorem
Bulgarian Academy of Sciences - National Committee for Mathematics
Mathematica Balkanica New Series, Vol. 27, Fasc 1-2 (2013), 77p-87p
The classical notion of apolarity is defined for two algebraic polynomials of equal degree. The main property of two apolar polynomials p and q is the classical Grace theorem: Every circular domain containing all zeros of p contains at least one zero of q and vice versa. In this paper, the definition of apolarity is extended to polynomials of different degree and an extension of the Grace theorem is proved. This leads to simplification of the conditions of several well-known results about apolarity.