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 $\nabla$ 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 $\&nabla$ intersects the boundary $\partial D$ along $n$ real algebraic surfaces $\mathscr S^{(j)}$ of dimension $n-2$. Furthermore, every branch $y_j(x)$ of the general algebraic function ramifies in $D$ only along the pair of surfaces $\mathscr S^{(j)}$ and $\mathscr S^{(j-1)}$.
Keywords:algebraic equation, hypergeometric function, discriminant, integral representation, monodromy.
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