A new counting function for the zeros of holomorphic curves

Abstract

Let $f_1,\ldots ,f_p$ be entire functions that do not all vanish at any point, so that $(f_1,\ldots ,f_p)$ is a holomorphic curve in ${\mathbb {C}}{\mathbb {P}}^{p-1}$. We introduce a new and more careful notion of counting the order of the zero of a linear combination of the functions $f_1,\ldots ,f_p$ at any point where such a linear combination vanishes, and, if all the $f_1,\ldots ,f_p$ are polynomials, also at infinity. This enables us to formulate an inequality, which sometimes holds as an identity, that sharpens the classical results of Cartan and others.