Introducing isodynamic points for binary forms and their ratios

TL;DR

This paper generalizes the concept of isodynamic points from triangles to polynomials and binary forms, introducing an isodynamic map that remains invariant under Mobius transformations, revealing new geometric and algebraic invariants.

Contribution

It introduces a novel isodynamic map for polynomials and binary forms that generalizes classical triangle invariants to algebraic curves and their ratios.

Findings

The isodynamic map commutes with Mobius group actions.

Roots of the image polynomial define the isodynamic points.

Extension from univariate polynomials to binary forms and 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 Mobius group on the set of triangles. Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree d at least 3 with at most double roots a polynomial of degree (at most) 2d-4 such that this map commutes with the action of the Mobius group 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.