E-functions are entire functions whose Taylor coefficients at 0 behave both from the analytic and the arithmetic viewpoints as those of the exponential function. Let $f_1, ..., f_n$ be a vector of E-functions, algebraically independent over $Q(z)$, satisfying a differential system $dY/dz = A(z) Y$ , where the entries of $A$ are rational functions with algebraic coefficients. Let further $b$ be an algebraic number outside the poles of $A$. The theorem of Siegel and Shidlovsky, which generalizes the Lindemann - Weierstrass theorem, then asserts that the numbers $f_1(b),..., f_n(b)$ are algebraically independent over $Q$.
Recently, Yves Andr\'e found an entirely new proof of this classical result. Although based on deep results from the arithmetic theory of differential equations, its principle is remarkably simple. By a Fourier transformation, one deduces from the Katz-Honda theory in characteristic p , Chudnovsky's theorem on $G$-operators, and Andre's theorem on their p-adic radii of convergence, that the minimal differential operator annihilating an $E$-function $F$ has large exponents at any algebraic zero of large order of $F$. The result then follows from an extension of Fuchs' global relation on exponents to the case of irregular singularities.