Dans ce travail nous développons un analogue relatif de la théorie de Sen pour les $B_{dR}$-représentations. On donne des applications à la théorie des représentations $p$-adiques, en la reliant à la théorie des $(\varphi ,\Gamma )$-modules relatifs, et à celle des modules de Higgs $p$-adiques développée par G. Faltings.

The differential automorphism group, over F, Π₁(F₁) of the Picard-Vessiot closure F₁ of a differential field F is a proalgebraic group over the field $C_F$ of constants of F, which is assumed to be algebraically closed of characteristic zero, and its category of $C_F$ modules is equivalent to the category of differential modules over F. We show how this group and the category equivalence behave under a differential extension E ⊃ F, where $C_E$ is also algebraically closed.

One of the classical Bernstein inequalities compares the maxima of a polynomial of a given degree on the interval [-1,1] and on the ellipse in the complex plane with the focuses -1, 1 and the semiaxes $R$. We prove a similar inequality for a branch of an algebraic function of a given degree on the maximal disk of its regularity, with the explicitly given constant, depending on the degree only. In particular, this improves a recent inequality of Fefferman and Narasimhan and answers one of their questions....

Border bases are an alternative to Gröbner bases. The former have several more desirable properties. In this paper some constructions and operations on border bases are presented. Namely; the case of a restriction of an ideal to a polynomial ring (in a smaller number of variables), the case of the intersection of two ideals, and the case of the kernel of a homomorphism of polynomial rings. These constructions are applied to the ideal of relations and to factorizable derivations.