Introducing isodynamic points for binary forms and their ratios

Abstract

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