Newton Polygon

This paper is an introduction to the use of techniques originally developed by Newton to study local solutions of algebraic and ordinary differential equations. We first study the application of Newton’s polygon to algebraic equations with coefficients in a valued field, showing the limitations of its use for valuations of rank greater than one. The Theorem of Kaplansky is the key to have explicit solutions for the rank one case. For differential equations, the lack of a Differential Kaplansky’s theorem, is an obstacle to have a general theory, but we study ordinary differential equations with generalized series with exponents in subgroups of \(\mathbb {R}\) as coefficients. In booth cases we pointed up the possibility of the use valuations as solution of equations. The author wishes to thank to the referee for his (many) precise and detailed corrections that have improved this paper.