2017/06/19 by Qiming Yan, Yan, Qiming, Guangsheng Yu +1
Mathematics · #30D35 #32H30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1706.05896
openalex publication_date 2017/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f be a holomorphic curve in ℙn(ℂ) and let D=\D1,…,Dq\ be a family of moving hypersurfaces defined by a set of homogeneous polynomials Q=\Q1,…,Qq\. For j=1,…,q, denote by Qj=∑i0+⋯+in=djaj,I(z)x0i0⋯ xnin, where I=(i0,…,in)∈ℤ≥ 0n+1 and aj,I(z) are entire functions on ℂ without common zeros. Let KQ be the smallest subfield of meromorphic function field M which contains ℂ and all \fracaj,I'(z)aj,I''(z) with aj,I''(z)\not≡ 0, 1≤ j≤ q. In previous known second main theorems for f and D, f is usually assumed to be algebraically nondegenerate over KQ. In this paper, we prove a second main theorem in which f is only assumed to be nonconstant. This result can be regarded as a generalization of Cartan's conjecture for moving hypersurfaces.