Bicomplex polar weighted homogeneous polynomials

2025/05/30 by Yesenia Bravo, Bravo, Yesenia, Inácio Rabelo +3
Computer Science · Mathematics · #14B05 #30G35 #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Primary: 32S55 #Secondary: 32C18

paper · pdf · doi:10.48550/arxiv.2506.00255

openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the topology of real polynomial maps ℝ⁴ⁿ \longrightarrow ℝ⁴ 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.

Citations

• Morphisms of Cartan connections
• Topology of polar weighted homogeneous hypersurfaces
• Refinements of Milnor's Fibration Theorem for Complex Singularities
• Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici

Related

• Topology of the Milnor fibrations of polar weighted homogeneous polynomials
• Topology of polar weighted homogeneous hypersurfaces
• Results on Milnor Fibrations for mixed polynomials with non-isolated singularities
• Fold singularities of the maps associated with Milnor fibration for mixed polynomials
• On Mixed Brieskorn Variety