A uniqueness theorem with moving targets without counting multiplicity (Q2716114)

In 1926 R. Nevanlinna proved that two nonconstant meromorphic functions of a complex variable which take five distinct values at the same points, i.e. \(\exists\) distinct \(a_1,a_2,\dots,a_5\) such that NEWLINE\[NEWLINEf(x)=a_j \Leftrightarrow g(x)=a_j \quad(j=1,2, \dots,5),NEWLINE\]NEWLINE must be identical. In 1989 \textit{W. Stoll} extended this result to holomorphic curves [Pac. J. Math. 139, 311-337 (1989; Zbl 0683.32017)]. The present paper generalizes Stoll's result.NEWLINENEWLINENEWLINEFor a holomorphic curve \(f:\mathbb{C}\to \mathbb{P}^n(\mathbb{C})\), we use \({\mathbf f}:\mathbb{C} \to\mathbb{C}^{n+1} -\{0\}\) to denote a reduced representation of \(f\), that is \(\mathbb{P}({\mathbf f})=f\). We note that a hyperplane \(H\) in \(\mathbb{P}^n (\mathbb{C})\) can be regarded as a point in \(\mathbb{P}^n(\mathbb{C}^*)\), where \(\mathbb{C}^*\) is the dual space of \(\mathbb{C}\). By a moving target we mean a holomorphic map \(g:\mathbb{C} \to\mathbb{P}^n (\mathbb{C}^*)\). Let \({\mathbf g}:\mathbb{C} \to\mathbb{C}^{*n+1} -\{0\}\) be a reduced representation of \(g\). Then \({\mathbf g}({\mathbf f})\) is an entire function on \(\mathbb{C}\). Note that although the function \({\mathbf g}({\mathbf f})\) depends on the choice of representations, the zeros of \({\mathbf g}({\mathbf f})\) however are independent of the choice of representations. Moving targets \(g_j:\mathbb{C}\to \mathbb{P}^n (\mathbb{C}^*)\), \(1\leq j\leq q\), are said to be in general position if there is \(z_0\in\mathbb{C}\) such that the hyperplanes \(g_j(z_0)\), \(1\leq j\leq q\), are located in general position. Let \(f_1,f_2, \dots,f_\lambda: \mathbb{C}\to \mathbb{P}^n (\mathbb{C})\) be non-constant holomorphic curves. Let \(g_j:\_bbfC \to\mathbb{P}^n (\mathbb{C}^*) \), \(1\leq j\leq q\), be moving targets located in general position. Assume that \({\mathbf g}_j({\mathbf f}_t)\not \equiv 0\) for \(1\leq j\leq q\), \(1\leq t\leq \lambda\), and assume that \(({\mathbf g}_j({\mathbf f}_1))^{-1}\{0\} =\cdots =({\mathbf g}_j( {\mathbf f}_\lambda))^{-1} \{0\}\). Let \(A_j=({\mathbf g}_j({\mathbf f}_1))^{-1} \{0 \}\). Denote by \(T[n+1,q]\) the set of all injective maps from \(\{1,\dots, n+1\}\) to \(\{1,\dots,q\}\). For every \(z\in\mathbb{C} -\bigcup_{\beta\in T[n+1,q]} \{z\mid {\mathbf g}_{\beta(1)} (z)\wedge\cdots \wedge{\mathbf g}_{\beta(n+1)} (z)=0\}\), we define \(\rho(z)= \#\{j\mid z\in A_j\}\). Then \(\rho(z)\leq n\). For any positive number \(r>0\), define \(\rho(r)=\sup\{\rho(z)\mid|z|\leq r\}\) where the sup is taken over all \(z\in\mathbb{C}-\bigcup_{\beta\in T[n+1,q]}\{z\mid{\mathbf g}_{\beta (1)}(z)\wedge \cdots\wedge {\mathbf g}_{\beta (n+1)}(z)=0\}\) with \(|z|\leq r\). Then \(\rho(r)\) is a decreasing function. Let \(d=\lim_{r\to \infty}\rho (r)\). Then \(d\leq n\). If for each \(i\neq j\), \(A_i\cap A_j=\emptyset\), then \(d=1\). A number \(r_0>1\) exists such that NEWLINE\[NEWLINE\rho(z)\leq d\leq n,\text{ if }|z|\geq r_0\text{ and }z\notin \bigcup_{\beta\in T[n+1,q]} \bigl\{z\mid{\mathbf g}_{\beta(1)} (z)\wedge \cdots\wedge {\mathbf g}_{\beta (n+1)}(z)=0\bigr\}.NEWLINE\]NEWLINE The following result (a.o.) is obtained in this paper.NEWLINENEWLINENEWLINETheorem 1. Let \(f_1,f_2,\dots, f_\lambda: \mathbb{C}\to \mathbb{P}^n (\mathbb{C})\) be non-constant holomorphic curves. Let \(g_j:\mathbb{C} \to\mathbb{P}^n (\mathbb{C}^*)\) be moving targets located in general position and \(T_{g_j}(r)=o (\max_{1\leq t\leq\lambda} \{T_{f_t}(r)\})\), \(1\leq j\leq q\). Assume that \({\mathbf g}_j({\mathbf f}_t)\not \equiv 0\) for \(1\leq j\leq q\), \(1\leq t \leq \lambda\), and \(({\mathbf g}_j({\mathbf f}_1))^{-1} \{0\}=\cdots =({\mathbf g}_j ({\mathbf f}_\lambda))^{-1} \{0\}\). Denote by \(A_j=({\mathbf g}_j({\mathbf f}_1))^{-1}\{0\}\) and let \(A=\bigcup^q_{j=1} A_j\). Let \(l\), \(2\leq l\leq\lambda\), be an integer such that for any increasing sequence \(1\leq j_1<j_2<\cdots <j_l\leq \lambda\), \({\mathbf f}_{j_1}(z) \wedge\cdots \wedge{\mathbf f}_{j_t} (z)=0\) for every point \(z\in A\). If \(q>{dn^2(2n+1) \lambda\over \lambda-l+1}\), then \(f_1,\dots, f_\lambda\) are algebraically dependent over \(\mathbb{C}\), i.e. \({\mathbf f}_1\wedge\cdots \wedge{\mathbf f}_\lambda \equiv 0\) on \(\mathbb{C}\).