Puiseaux's Theorem

The whole neighborhood of any point $y_i$ of an algebraic Plane Curve may be uniformly represented by a certain finite number of convergent developments in Power Series,

$$x_i=\rho_\nu y_i+a_{\nu i1}t_\nu+a_{\nu i2}{t_\nu}^2+\ldots.$$

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 207, 1959.