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.
If 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 ? We begin with a naive heuristic. There are possibilities for , and for these there are values taken by . If the values taken by were ‘random’, then (1) would have roughly integer solutions , for large .
A lot of the time this is actually true in a precise sense. What can go wrong? If is even and the coefficients all have the same sign, then (1) admits only the trivial solution, since it does not even have nontrivial real solutions. If
then cannot have nontrivial zeros, by infinite descent (any zero has 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 -adic solubility for all primes (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 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 to (1). This asymptotic formula, as well as being a constant times , is the product of the local densities of solutions to (1). This is often called a quantitative Hasse principle.