Differential algebra aims at studying differential equations from a purely algebraic point of view. We can consider J. Ritt as the father of differential algebra.This thesis is based on the article ”Computing representations for radicals of finitely generated differential ideals” written by F. Boulier, D. Lazard, F. Ollivier and M.Petitot in 2009. We describe an algorithm, named Rosenfeld–Gr?bner, which computes a representation for the radical p of the differential ideal generated by any such system ?. This algorithm represents the radical of the differential ideal generated by ? ofpolynomial differential equations as a finite intersection of differential ideals ? ’s (that we call regular) The Rosenfeld–Gr?bner algorithm relies mainly on three theorems: · A theorem of zeros, which states that a polynomial p belongs to the radical of an ideal presented by a basis ? if and only if every solution of ? is a solution of p; we apply this theorem in the algebraic and in the differential case, · A lemma of Rosenfeld, which gives a sufficient condition so that a system of polynomial differential equations admits a solution if and only if this same system, considered as a purely algebraic system admits a solution, · A lemma of Lazard, which establishes that each regular ideal ? is radical and that all its prime components have a same parametric set (this property is stronger than “defining an unmixed algebraic variety”). In this thesis we prove Lazard’s Lemma and show how some computations can be performed in dimension zero. Chapter 3 contains differential algebra preliminaries. In continue we prove our version of Rosenfeld’s Lemma and some technical results which will be used for efficiently testing the coherence hypothesis of this lemma (in particular, we show there our analog of Buchberger’s second criterion). In chapter 4 we prove a result to represent radical differential ideals as intersections of regular differential ideals. This is the core of the Rosenfeld–Gr?bner algorithm. We present an algorithm to compute canonical representatives for regular differential ideals and state the Rosenfeld–Gr?bner algorithm as a theorem with an effective proof. We exploit also the use of the differential form of Buchberger's criteria. We explain how algebraic solutions of regular differential ideals can be expanded as formal power series. A few examples are developed. Finally, we present a generalization of the differential form of Buchberger's first criterion, and see that it can enhances the performance of the algorithm. We explain that the algebraic solutions of regular differential ideals can be expanded as formal power series. A few examples are developed in the last section. Keywords: Differential Algebra, differential equations with partial derivatives, differential ideal, Rosenfeld–Gr?bner algorithm, theorem of zeros, Rosenfeld lemma, Lazard lemma