The base points of the system of polar curves of an irreducible algebroid plane curve with general moduli are determined. As consequences a lower bound for the Tjurina number and many continuous analytic invariants of the curve are found.
One of the classical Bernstein inequalities compares the maxima of a polynomial of a given degree on the interval [-1,1] and on the ellipse in the complex plane with the focuses -1, 1 and the semiaxes . We prove a similar inequality for a branch of an algebraic function of a given degree on the maximal disk of its regularity, with the explicitly given constant, depending on the degree only. In particular, this improves a recent inequality of Fefferman and Narasimhan and answers one of their questions....