We generalise Birch's seminal work on forms in many variables to handle a system of forms in which the degrees need not all be the same. This allows us to prove the Hasse principle, weak approximation, and the Manin-Peyre conjecture for a smooth and geometrically integral variety X Pm, provided only that its dimension is large enough in terms of its degree.
- Complete intersections
- Forms in many variables
- Hardy-Littlewood circle method
- Hasse principle
- Rational points Manin conjecture
- Weak approximation