### A criterion for the existence of a flat connection on a parabolic vector bundle.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We consider polynomial mappings (f,g) of ℂ² with constant nontrivial jacobian. Using the Riemann-Hurwitz relation we prove among other things the following: If g - c (resp. f - c) has at most two branches at infinity for infinitely many numbers c or if f (resp. g) is proper on the level set ${g}^{-1}\left(0\right)$ (resp. ${f}^{-1}\left(0\right)$), then (f,g) is bijective.

In this paper we give a new algebro-geometric proof to the semi-group theorem due to Abhyankar-Moh for the affine plane curves with one place at infinity and its inverse theorem due to Sathaye-Stenerson. The relations between various invariants of these curves are also explained geometrically. Our new proof gives an algorithm to classify the affine plane curves with one place at infinity with given genus by computer.

We give an upper bound for the order of an automorphism of a Riemann surface with two fixed points. The main results are presented in Theorems 1.4 and 2.4.