We recall Labatie's effective method of solving polynomial equations with two unknowns by using the Euclidean algorithm.

Let F(X, Y ), G(X, Y ) be polynomials of degrees m, n > 0 respectively. We prove, that the set {(x, y) Є R² : F(x, y) = G(x, y) = 0} has at most mn connected components.

We provide the detailed proof of a sharpened version of the M. Artin Approximation Theorem.

We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is ...

We consider the algebroid plane curves de ned by formal power series of two variables with coe cients in an algebraically closed eld. Using quadratic transformations we prove the local normalization theorem. Then we study ...