  E-functions are entire functions with algebraic Taylor coefficients
satisfying certain arithmetic conditions, and which are also solutions of
linear differential equations with polynomial coefficients. They were
introduced by Siegel in 1929 to generalize the Diophantine properties of the
exponential and Bessel's functions. The Siegel-Shidlovskii Theorem (1956) deals
with the algebraic (in)dependence of values at algebraicpoints of E-functions
solutions of a differential system. In this paper, we prove the existence of an
algorithm to perfom the following three tasks. Given as inputs some E-functions
$F_1(z), ..., F_p(z)$, (1) it computes a system of generators of the ideal of
polynomial relations between $F_1(z), ..., F_p(z)$; (2) given any algebraic
number $\alpha$, it computes a system of generators of the ideal of polynomial
relations between the values $F_1(\alpha), ..., F_p(\alpha)$ with algebraic
coefficients;(3) if $z,F_1(z), ..., F_p(z)$ are algebraically independent, it
determines the finite set of all algebraic numbers $\alpha$ such that the
values $F_1(\alpha), ..., F_p(\alpha)$ are algebraically dependent. The
existence of this algorithm relies on a variant of the Hrushovski-Feng
algorithm (to compute polynomial relations between solutions of differential
systems) and on Beukers' lifting theorem (an optimal refinement of the
Siegel-Shidlovskii theorem) in order to reduce the problem to an effective
elimination procedure in multivariate polynomial rings. The latter is then
performed using Gr{\"o}bner bases.
