Topology of polar weighted homogeneous hypersurfaces

TL;DR

This paper investigates the topological properties of hypersurfaces defined by polar weighted homogeneous polynomials, revealing their similarities to complex weighted homogeneous polynomials and expanding understanding of their geometric structure.

Contribution

It introduces the basic properties of hypersurfaces defined by polar weighted homogeneous polynomials, highlighting their topological and geometric features.

Findings

Polar weighted homogeneous hypersurfaces share properties with complex weighted homogeneous ones.

The paper establishes foundational topological characteristics of these hypersurfaces.

It provides a framework for analyzing real polynomial hypersurfaces with polar symmetry.

Abstract

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.