Introductory post

# Introduction to diophantine equations in many variables

Suppose we want to study the integer zeros of a polynomial in s-1 variables. Equivalently we can homogenise and consider a form in s variables. Let’s start with the simplest case: an additive form with nonzero integer coefficients.

$F(\mathbf{x}) = a_1 x_1^k + \ldots + a_s x_s^k \qquad \qquad (1)$

If $s$ is small, then the problem can be attacked arithmetically. We shall be concerned with the situation where there are many variables. How many solutions do we expect in a box of length $B$? We begin with a naive heuristic. There are $\asymp B^s$ possibilities for $\mathbf{x}$, and for these $\mathbf{x}$ there are $\asymp B^k$ values taken by $F$. If the values taken by $F$ were ‘random’, then (1) would have roughly $B^{s-k}$ integer solutions $\mathbf{x} \in [-B,B]^s$, for large $B$.

A lot of the time this is actually true in a precise sense. What can go wrong? If $k$ is even and the coefficients $a_1, \ldots, a_s$ all have the same sign, then (1) admits only the trivial solution, since it does not even have nontrivial real solutions. If

$F(x,y,z) = x^5 + 3y^5 + 9z^5 \qquad \qquad (1)$

then $F$ cannot have nontrivial zeros, by infinite descent (any zero has $x,y,z$ all divisible by 3, which rules out the possibility of a smallest nontrivial zero). See the introduction of this for more examples.

The examples above illustrate failures of real solubility and 3-adic solubility, respectively. Given real and $p$-adic solubility for all primes $p$ (collectively referred to as local solubility), we might expect (1) to admit a nontrivial solution. This philosophy is known as a Hasse principle, or local-global principle. It’s not always true, but if $s$ is large then it holds under mild conditions.

In fact much more can be said. In a future post, I’ll explain how the circle method can provide an asymptotic formula for the number of integer solutions $\mathbf{x} \in [-B,B]^s$ to (1). This asymptotic formula, as well as being a constant times $B^{s-k}$, is the product of the local densities of solutions to (1). This is often called a quantitative Hasse principle.