Forms of differing degrees over number fields

Consider a system of m forms of degree d in n variables over the
integers. The classical question on the number of integers solutions
was solved by Birch. Using the circle method he gave an asymptotic
formula for the number of integer solutions to this system in a
homogeneously expanding box provided n is large compared to m and
d. An analogous result over arbitrary number fields was proved by
Skinner, where the asymptotic formula is independent of the degree of
the number field. In joint work with C.~Frei, we combine Skinner's
techniques with a recent generalization of Birch's theorem by Browning
and Heath-Brown, where they allow the forms to have differing degrees.

We discuss the main ingredients of the proof as well as consequences
of this result for the Hasse principle, weak approximation and Manin's
conjecture.