Abstract
We consider a general reduced algebraic equation of degree n with complex coefficients. The solution to this equation, a multifunction, is called a general algebraic function. In the coefficient space we consider the discriminant set ∇ of the equation and choose in its complement the maximal polydisk domain D containing the origin. We describe the monodromy of the general algebraic function in a neighborhood of D. In particular, we prove that ∇ intersects the boundary ∂D along n real algebraic surfaces \(S^{(j)} \) of dimension n − 2. Furthermore, every branch y j (x) of the general algebraic function ramifies in D only along the pair of surfaces \(S^{(j)} \) and \(S^{(j - 1)} \).
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The author was supported by the Russian Foundation for Basic Research (Grant Grant 14-01-31265-mol_a) and by the Grant of the Russian Federation for the State Support of Leading Researches of the Siberian Federal University (Agreement 14.Y26.31.0006).
Krasnoyarsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 2, pp. 409–419, March–April, 2015.
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Mikhalkin, E.N. The monodromy of a general algebraic function. Sib Math J 56, 330–338 (2015). https://doi.org/10.1134/S0037446615020123
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DOI: https://doi.org/10.1134/S0037446615020123