  A small value estimate is a statement providing necessary conditions for the
existence of certain sequences of non-zero polynomials with integer
coefficients taking small values at points of an algebraic group. Such
statements are desirable for applications to transcendental number theory to
analyze the outcome of the construction of an auxiliary function. In this
paper, we present a result of this type for the product Ga x Gm whose
underlying group of complex points is C x C*. It shows that if a certain
sequence of non-zero polynomials in Z[X,Y] take small values at a point
(xi,eta) together with their first derivatives with respect to the invariant
derivation of the group, then both xi and eta are algebraic over Q. The precise
statement involves growth conditions on the degree and norm of these
polynomials as well as on the absolute values of their derivatives. It improves
on a direct application of Philippon's criterion for algebraic independence and
compares favorably with constructions coming from Dirichlet's box principle.
