2023/06/05 by Benjamin Bode, Bode, Benjamin, Eder L. Sanchez Quiceno +1 · 1 citation
Mathematics · #14B05 #14J17 #14M25 #14P05 #32S05 #32S55 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2306.02905
openalex publication_date 2023/06/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Mixed polynomials f:ℂ2→ℂ are polynomial maps in complex variables u and v as well as their complex conjugates u and v. They are therefore identical to the set of real polynomial maps from ℝ4 to ℝ2. We generalize Mondal's notion of partial non-degeneracy from holomorphic polynomials to mixed polynomials, introducing the concepts of partially non-degenerate and strongly partially non-degenerate mixed functions. We prove that partial non-degeneracy implies the existence of a weakly isolated singularity, while strong partial non-degeneracy implies an isolated singularity. We also compare (strong) partial non-degeneracy with other types of non-degeneracy of mixed functions, such as (strong) inner non-degeneracy, and find that, in contrast to the holomorphic setting, the different properties are not equivalent for mixed polynomials. We then introduce additional conditions under which strong partial non-degeneracy becomes equivalent to the existence of an isolated singularity. Furthermore, we prove that mixed polynomials that are strongly inner non-degenerate satisfy the strong Milnor condition, resulting in an explicit Milnor (sphere) fibration.