On the Number of Linearly Independent Solutions of the Riemann Boundary Value Problem on the Riemann Surface of an Algebraic Function
Tóm tắt
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.
Tài liệu tham khảo
Zverovich E.I. "Boundary value problems in the theory of analytic functions in Hoelder classes on Riemann surfaces", Russian Math. Surveys 26 (1), 117-192 (1971).
Chebotarev N.G. Theory of algebraic functions (OGIZ, Moscow–Leningrad, 1948) [in Russian].
Kruglov V.E. "Abelian differentials and equations of surface given by a cyclic permutation group", Soobsch. AN Gruzinskoi SSR 92 (3), 537-540 (1978) [in Russian].
Kruglov V.E. "Partial indexes, abelian differentials of the first kind and the equation of a surface given by a finite abelian group of permutations", Siberian Math. J. 22 (6), 872-882 (1981).
Kruglov V.E. "Partial indices and an application of factorization of some matrices of permutation type at most fourth order", Siberian Math. J. 24 (2), 200-201 (1983) (Dep. VINITI No.3278-82 Dep., No.3279-82 Dep., No.3280-82 Dep) [in Russian].
Shtin S.L. Constructing basic functionals of Riemann surfaces and the V.E. Kruglov conjecture on groups of permutation matrices, Dissertation kand. phys.-math. nauk (Minsk, 1998) [in Russian].
Vekua N.P. Systems of singular integral equations (Nauka, Moscow, 1970) [in Russian].
Kruglov V.E. "Structure of the partial indices of the Riemann problem with permutation-type matrices", Math. Notes 35 (2), 89-93 (1984).
Hyrwitz A., Courant R. Function Theory (Nauka, Moscow, 1968) [in Russian].
Hensel K., Landsberg G. Theorie der abelschen Funktionen einer Variablen (Lpz, Teubner, 1902).
Kruglov V.E. "On properties of Hensel–Landsberg kernel on Riemann surfaces of algebraic functions", in: Theory of functions of a complex variables and boundary value problems, 78-87 (Cheboksary, 1974) [in Russian].
Kruglov V.E. "On algebraic functions divisible by a given divisor", Dokl. Math. 44 (3), 647-649 (1991).
Kruglov V.E. "Factorization of matrices of permutation type", Ukrainian Math. J. 46 (11), 1627-1633 (1994).
Kruglov V.E. "Solution of Riemann boundary value problem on an n-sheeted Riemann surface", Matem. Issledovaniya 9 (2), 230-236 (1974) [in Russian].