A new counting function for the zeros of holomorphic curves (Q461361)

Let \((f_1,\ldots,f_n)\) be a holomorphic curve of \(\mathbb{C}\) into \(\mathbb{P}^{n-1}(\mathbb{C})\). The authors introduce a new way of counting the order of the zero of a linear combination of \(f_1,\ldots,f_n\) in the complex plane. If all functions \(f_1,\ldots,f_n\) are polynomials, the method introduced by the authors is also applicable at infinity. Using this more careful way of counting zeros, the authors formulate an inequality, which improves the classical theorem by \textit{H. Cartan} [Mathematica, Cluj 7, 5--31 (1933; Zbl 0007.41503)].