On the degree of Polar Transformations -- An approach through Logarithmic Foliations

TL;DR

This paper studies the degree of polar transformations linked to homogeneous functions, revealing that it depends on the zero locus of the defining polynomial, using algebraic geometry and logarithmic foliations.

Contribution

It provides an algebro-geometric proof relating the degree of polar transformations to the zero locus of the polynomial, extending previous topological results.

Findings

Degree of polar transformation pre-image determined by zero locus of polynomial

Established algebraic geometric methods for analyzing polar transformations

Extended results to higher-dimensional linear spaces

Abstract

We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation associated to a homogeneous polynomial $F$ is determined by the zero locus of $F$. For zero dimensional-dimensional linear spaces this was conjecture by Dolgachev and proved by Dimca-Papadima using topological arguments. Our methods are algebro-geometric and rely on the study of the Gauss map of naturally associated logarithmic foliations.