Divisors and Differential Forms

Abstract

A polynomial in one variable is uniquely determined to within a constant factor by its roots and their multiplicities, that is, by a collection of points x₁, …, x_r ∈ A¹ with multiplicities l₁, …, l_r. A rational function ϕ(x) = f(x)/g(x), f, g ∈ k[X] is determined by the zeros of the polynomials f and g, that is, by the points at which it vanishes or is non-regular. To distinguish the roots of g from those off we take their multiplicities with a minus sign. Thus, ϕ is given by points x₁, …, x_r with arbitrary integral multiplicities l₁, …, l_r. Now we set ourselves the task of specifying a rational function on an arbitrary algebraic variety in a similar way.