Boundaries of holomorphic 1-chains within holomorphic line bundles over ℂℙ ¹

We show that boundaries of holomorphic 1-chains within holomorphic line bundles of ℂℙ ¹ can be characterized using a single generating function of Wermer moments. In the case of negative line bundles, a rationality condition on the generating function plus the vanishing moment condition together form an equivalent condition for bounding. We provide some examples which reveal that the vanishing moment condition is not sufficient by itself. These examples also can be used to demonstrate one point of caution about the use of birational maps in this topic. In the case of positive line bundles, where the vanishing moment condition vacuously holds, boundaries of holomorphic 1-chains can be characterized using the aforementioned rationality condition modulo a series of polynomial terms whose degrees are dependent on the degree of the line bundle. As a side point with potential independent interest, we show for any meromorphic function that rationality with prescribed bounds on degree is equivalent to the satisfaction of a particular determinantal differential equation.