Computer Algebra and Bifurcations

In this paper we will present the family of Newton algorithms. From the computer algebra point of view, the most basic of them is well known for the local analysis of plane algebraic curves f(x,y)=0 and consists in expanding y as Puiseux series in the variable x. A similar algorithm has been developped for multi-variate algebraic equations and for linear differential equations, using the same basic tools: a “regular” case, associated with a “simple” class of solutions, and a “simple” method of calculus of these solutions; a Newton polygon; changes of variable of type ramification; changes of unknown function of two types y=ctμ+ϕ or y=exp (c/tμ)ϕ.