On the number of linearly independent solutions of the Riemann boundary value problem on the Riemann surface of an algebraic function

V. E. Kruglov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2021), pp. 15-36.

Résumé

In this paper, a modified solution of the Riemann boundary value problem on a Riemann surface (R.S.) of an algebraic function kind $\rho$ is proposed. This allowed finding the number l of linearly independent algebraic functions (LIAF), that are multiples of a fractional divisor $Q$, to reduce to finding the number of LIAF that are multiples of an integer divisor $J$ (${\rm ord}\, J = \rho$). It provides a solution of the Jacobi inversion problem obtained in this paper. In this paper, we study the case when the exponents of the normal basis elements coincide, and the problem of finding the number of LIAF, multiples of an integer divisor, is solved. The definitions of conjugate points of R.S. and a hyperorder of a whole divisor are introduced. Depending on the structure of the divisor $J$, exact formulae are obtained for the number $l$, expressed in terms of the divisor $Q$ order, the hyperorder of the divisor $J$, and the numbers $\rho$ and $n$, where $n$ is the number of sheets of algebraic function R.S.

Détail

Keywords: Riemann boundary value problem, Riemann surface of an algebraic function, rank of a matrix.

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