This PhD thesis is mostly devoted to the computation of p-adic lifting by relaxed algorithms. In a first part, we introduce relaxed algorithms and their application to the computation of recursive p-adics. In order to use this framework for the p-adic lifting of various systems of equations, we have to transform the given implicit equations into recursive equations. The case of systems of linear equations, possibly differential, is treated in the second part. This third part contains the lifting of resolutions of polynomial systems. In any cases, these new relaxed algorithms are compared, both in theory and practice, to existing algorithms. In the fourth part, we focus on the universal decomposition algebra. We present a fast algorithm which computes an adequate representation of this algebra and use it to compute efficiently with the elements of this algebra. Finally, we show in the appendix that finding fundamental invariants of polynomial invariants algebras under a finite group can be done directly modulo p, hence making their computation easier.
