A new counting function for the zeros of holomorphic curves

Let f(1),..., f(p) be entire functions that do not all vanish at any point, so that (f(1),..., f(p)) is a holomorphic curve in CPp-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),..., f(p) at any point where such a linear combination vanishes, and, if all the f(1),..., 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.