A family of polynomials with the uniqueness property for linearly non-degenerate holomorphic mappings (Q1872586)

Let \(F\) be a family of nonconstant holomorphic mappings of \(\mathbb{C}\) into \(P^n(\mathbb{C})\) and \(P\) be a homogeneous polynomial of \(n+1\) variables. The polynomial \(P\) has the uniqueness property for \(F\) if \(P(F)\not\equiv 0\) for any \(f\in F\), and \(P(\overline f)=\alpha P(\overline g)\) implies \(f=g\) for any two elements of \(f\) and \(g\) of \(F\), where \(\overline f\) and \(\overline g\) are reduced representations of \(f\) and \(g\), respectively, and \(\alpha\) is an entire function without zeros. The main result obtained here is: Suppose \(d\geq(2q+1)^2\). Then, the polynomial \(P\) has the uniqueness property for linearly non-degenerate holomorphic mappings of \(\mathbb{C}\) into \(P^n(\mathbb{C})\), that is, for linearly non-degenerate holomorphic mappings \(f\) and \(g\) of \(\mathbb{C}\) into \(P^n(\mathbb{C})\) with reduced representations \(f\) and \(g\), respectively, \[ \sum^q_{j=0} (v_j\cdot f)^d=\varphi \sum^q_{j=0} (v_j\cdot g)^d \] for an entire function \(\varphi\) without zeros implies \(f=g\).