Polars of real singular curves

TL;DR

This paper reviews classical and reciprocal polar varieties, focusing on real affine plane curves, and provides conditions under which these varieties contain points on all real components of singular curves.

Contribution

It introduces the concept of reciprocal polar varieties and analyzes their role in locating points on all real components of singular algebraic curves.

Findings

Conditions for polar varieties to contain points on all real components

Extension of classical polar varieties to reciprocal polar varieties

Application to real affine plane curves with singularities

Abstract

Polar varieties have in recent years been used by Bank, Giusti, Heintz, Mbakop, and Pardo, and by Safey El Din and Schost, to find efficient procedures for determining points on all real components of a given non-singular algebraic variety. In this note we review the classical notion of polars and polar varieties, as well as the construction of what we here call reciprocal polar varieties. In particular we consider the case of real affine plane curves, and we give conditions for when the polar varieties of singular curves contain points on all real components.