  We study almost prime solutions of systems of Diophantine equations in the
Birch setting. Previous work shows that there exist integer solutions of size B
with each component having no prime divisors below $B^{1/u}$, where
$u=c_0n^{3/2}$, $n$ is the number of variables and $c_0$ is a constant
depending on the degree and the number of equations. We improve the polynomial
growth $$n^{3/2}$$ to the logarithmic $$\frac{\log n}{\log \log n}.$$ Our main
new ingredients are the generalisation of the Br\"udern-Fouvry vector sieve in
any dimension and the incorporation of smooth weights into the Davenport-Birch
version of the circle method.
