Introducing isodynamic points for binary forms and their ratios

Abstract

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group {mathcal {M}} on the set of triangles, Kimberling (Encyclopedia of Triangle Centers, http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree dge 3 with at most double roots a polynomial of degree (at most) 2d-4 such that this map commutes with the action of the Möbius group {mathcal {M}} on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.