  By adopting a new approach to the analysis of the density of p-adic solutions
arising in applications of the circle method, we show that under modest
conditions the existence of non-trivial p-adic solutions suffices to establish
positivity of the singular series. This improves on earlier approaches due to
Davenport, Schmidt and others, and allows us to establish an asymptotic formula
for the number of simultaneous zeros of non-singular pairs of cubic forms in at
least 131 variables. As a by-product, we obtain a version of Hensel's Lemma for
linear spaces.
