On the global distance between two algebraic points on a curve

We prove diophantine inequalities involving various distances between two distinct algebraic points of an algebraic curve. These estimates may be viewed as extensions of classical Liouvilleʹs inequality. Our approach is based on a transcendental construction using algebraic functions. Next we apply our results to Hilbertʹs irreducibility Theorem and to some classes of diophantine equations in the circle of Rungeʹs method.