Bulletin (new Series) of the American Mathematical Society
نویسنده
چکیده
Broadly speaking, one of the primary tasks in the theory of compact Riemann surfaces is the construction of meromorphic functions. A powerful tool in this endeavor is the Riemann-Roch theorem which allows the construction of functions with partially prescribed zeros and poles, such that the number of prescribed poles exceeds the zeros by a fixed constant depending on the surface. To be more precise, choose a finite collection of points pi and integers ni. This data, called a divisor, is usually represented as a sum D = n1p1 +n2p2 + . . . . The sum of the ni is the degree of D, which is denoted by deg D. The vector space L(D) consists of 0 together with meromorphic functions which vanish to order at least ni at pi if ni ≥ 0, or which have poles of order at most −ni at pi otherwise. The Riemann-Roch theorem gives a formula for the dimension:
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BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
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