Bicomplex polar weighted homogeneous polynomials

TL;DR

This paper explores the topology of bicomplex mixed polynomials, introducing polar weighted homogeneity, and establishes Milnor fibrations and homotopy descriptions for these polynomials, extending classical complex polynomial results.

Contribution

It introduces the concept of polar weighted homogeneity for bicomplex polynomials and develops a bicomplex Milnor fibration theory, extending previous complex polynomial studies.

Findings

Existence of global and spherical Milnor fibrations for bicomplex polynomials

Introduction of polar weighted homogeneity concept

Homotopy type description of polynomial fibers

Abstract

We study the topology of real polynomial maps $\mathbb{R}^{4n} \longrightarrow \mathbb{R}^{4}$ expressed in terms of bicomplex variables and their conjugates, which we refer to as bicomplex mixed polynomials. We introduce the notion of polar weighted homogeneity, a property that generalizes the concept of weighted homogeneity in the complex setting. This leads to the existence of global and spherical Milnor fibrations. Moreover, we include a discussion on bicomplex vector calculus, a bicomplex holomorphic analogue of the Milnor fibration theorem, and a theorem of Join type that describes the homotopy type of the fibers of certain polynomials on separable variables. This extends previous works on mixed polynomials in complex variables and their conjugates.