# Systems of cubic forms in many variables

## TL;DR

This paper proves an asymptotic formula for counting integer solutions to systems of smooth cubic forms in many variables, significantly reducing the variable count needed compared to previous results.

## Contribution

It establishes an asymptotic count for solutions when the number of variables is at least 25 times the number of forms, improving upon the prior requirement of much larger variable counts.

## Key findings

- Asymptotic formula for solutions in systems with n ≥ 25R variables.
- Reduction of variable count needed from R^2 to linear in R.
- Application of Davenport's method to bound solutions of auxiliary inequalities.

## Abstract

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 can handle systems of forms in $O(R)$ 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.