Vanishing sums in function fields

Source abstract

Let k be a field of zero characteristic, and let F be a function field over k of genus g . We normalize each valuation v on F so that its order group consists of all rational integers, and for elements u 1 , …, u n of F , not all zero, we define the (projective) height as The sum formula on F shows that this is really a height on the projective space .