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.