The monodromy of a general algebraic function

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 yj(x) of the general algebraic function ramifies in D only along the pair of surfaces \(S^{(j)} \) and \(S^{(j - 1)} \).