Abstract
We suggest a modified solution to the Riemann boundary value problem on a Riemann surface of an algebraic function of genus \(\rho\). This allows us to to reduce the problem of finding the number l of linearly independent algebraic functions (LIAF), that are multiples of a fractional divisor Q, to finding the number of LIAF that are multiples of an effective divisor J (\(\operatorname{ord}\,J = \rho\)); this provides a solution to the Jacobi inversion problem given in this paper. We study the case, where the exponents of the normal basis elements coincide, and solve the problem of finding the number of LIAF, multiples of an effective divisor. The definitions of conjugate points of Riemann surface and hyperorder of an effective divisor are introduced. Depending on the structure of divisor J, exact formulas are obtained for number l; they are expressed in terms of the order of divisor Q, the hyperorder of divisor J, and numbers \(\rho\) and n, where n is the number of sheets of the algebraic Riemann surface.
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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 10, pp. 15–36.
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Kruglov, V.E. On the Number of Linearly Independent Solutions of the Riemann Boundary Value Problem on the Riemann Surface of an Algebraic Function. Russ Math. 65, 10–30 (2021). https://doi.org/10.3103/S1066369X21100029
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DOI: https://doi.org/10.3103/S1066369X21100029