2021/07/19 by Sachiko Saito, Saito, Sachiko, Kosei Takashimizu +1
Mathematics · #14P05 #32S45 #Algebraic Geometry (math.AG) #Analytic and geometric function theory #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2107.08691
openalex publication_date 2021/07/19 · openalex created_date 2021/08/02 · openalex updated_date 2026/07/28
A mixed polynomial f(\boldsymbolz, \boldsymbolz) is called a mixed weighted homogeneous polynomial (Definition 5) if it is both radially and polar weighted homogeneous. Let f be a mixed weighted homogeneous polynomial with respect to a strictly positive radial weight vector P and a polar weight vector Q. Suppose that f is Newton non-degenerate over a compact face Δ(P) and polar weighted homogeneous of non-zero polar degree with respect to Q. Then f : ℂ^*n → ℂ has no mixed critical points. Moreover, under the assumption f-1(0) ∩ ℂ^*n ≠ ∅, f : ℂ^*n → ℂ is surjective. In other words, in this situation, Newton non-degeneracy over a compact face Δ(P) implies strong Newton non-degeneracy over Δ(P) (Proposition 10). With this fact as a starting point, we investigate the sets f-1(0) ∩ ℂ^*n, and show the existence of a collection of mixed weighted homogeneous polynomials f = fΔ(P) of non-zero polar degree which satisfy dim Δ(P) ≥ 1 and f-1(0) ∩ ℂ^*n = ∅ (Theorem 11). We also give an example of convenient mixed function germs of mixed weighted homogeneous face type which are not true non-degenerate (Definition 14).