The Circle Method

Hemar Godinho

Here we present a technique largely used today in diophantine problems, the Hardy-Littlewood Circle Method. The theory involved in this method is essentially analytic, and that makes it even more interesting, for it follows a ”modern ” tendency of our days of seeing mathematics as one body of knowledge. The method was first idealized to obtain an asymptotic formula for p(n), the number of partitions of n, where it proved to be very accurate. But its most famous applications are on the Goldbach and Waring problems, and great progress here was acheived thanks to the improvements on the method made by Vinogradov. Later on Davenport and Lewis extended it to work on simultaneous additive equations, followed by further developments due to Siegel and Radamacher that extended it to algebraic numbers.

This presentation begins with a very simpleminded overview of the method, followed by the proof of a result on pairs of rational quartic forms, that will be a hands-on illustration of the Circle method.