Systems of cubic forms in many variables

We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided {n\geq 25R}, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in {25R} variables, previous work having required that {n\gg R^{2}}. One conjectures that {n\geq 6R+1} should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.

Improvements in Birch’s theorem on forms in many variables

We show that a non-singular integral form of degree

On Fixed Divisors of Forms in Many Variables, I

Let $D_{d,r}$ be the maximal fixed divisor of a primitive form of degree $d$ in $r$ variables over $\mathsf{Z}$. A formula is given for $D_{d,2}$ and estimates for $D_{d,r}$ for $r>2$. As a consequence, a question of Nagell raised in 1919 is completely answered.