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Explicit construction of a global uniformization for an algebraic correspondence. (Russian, English) Zbl 1064.30037

Sib. Mat. Zh. 41, No. 1, 72-87 (2000); translation in Sib. Math. J. 41, No. 1, 61-73 (2000).
The global uniformization problem for an analytic correspondence \(f(z,w)=0\) is the problem of finding a way to pass from the implicit description \(f(z,w)=0\) to an equivalent parametric description \(z=\varphi (t)\), \(w=\psi (t)\), where \(\varphi\) and \(\psi\) are single-valued meromorphic functions in a parameter \(t\). The article is devoted to the problem of explicit construction of a global uniformization.
The authors consider the simplest case of a uniformization of an algebraic correspondence \(f(z,w)=0\). They start with an algorithm for decomposing the polynomial \(f(z,w)\) into irreducible factors. In particular, the question of irreducibility of a given polynomial is solved. The content of the corresponding section is at least of methodological interest as a new application of the Carleman “damping function.” The further exposition mainly follows Eh. I. Zverovich [Vestn. Beloruss. Gos. Univ. Im. V. I. Lenina, Ser. I 1991, No. 1, 36–39 (1991; Zbl 0773.30043)] and O. B. Dolgopolova, È. I. Zverovich [Boundary Value Problems, Spectral Functions and Fractional Calculus (in Russian), BGU, Minsk, 76-80 (1996)] but in more detail. In the final section, various examples of an explicit construction of a global uniformization are considered.

MSC:

30F10 Compact Riemann surfaces and uniformization
14H05 Algebraic functions and function fields in algebraic geometry

Citations:

Zbl 0773.30043
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