Three researchers from Bristol University are seeking to develop methods for analysing the distribution of integer solutions to polynomial equations.

How do you know when a polynomial equation has integer solutions?

Trevor Wooley, Tim Browning and Andy Booker from the mathematics department of Bristol University are using analytical methods to develop an algorithm which can be applied to certain classes of polynomials in order to determine the existence and distribution of integer solutions for any given equation.

In order to help visualise the patterns that occur, Ulrich Derenthal, at the University of Zurich, has plotted the possible solutions of small size to a given polynomial on a 3D computer-generated representation of the equation (below).

Booker, Browning and Wooley aim to encode polynomial solutions in a Dirichlet series, producing a means to predict the distribution of integer solutions.

The animation shows the integer solutions as red dots on the yellow 3D surface. The distribution of dots in clusters at the centre, and along the cone-like ‘arms’ of the surface, show patterns which the researchers aim to explain. It is conjectured that the geometry of the picture should govern the distribution of the solutions.

The study of polynomials has a long history stretching back to Diophantus in 3rd Century Alexandria. (Polynomial equations with integer coefficients are also known as Diophantine equations.)

Probably the most famous of these equations is the subject of Fermat’s Last Theorem:

xk + yk = zk,

for k>2. Another example is the Taxicab number, made famous by early 20th Century mathematicians GH Hardy and Ramanujan.

When Hardy was visiting the young Ramanujan in hospital, for want of something to say, Hardy remarked that he had travelled to the hospital in cab number 1729, which he described as “a rather dull number”. But Ramanujan pointed out that 1729 is the smallest number expressible as a sum of two cubes in two different ways:

13 + 123 = 93 + 103 = 1729,

which is a solution of the polynomial equation

x3 + y3 = u3 + v3.

Booker, Browning and Wooley are seeking to understand the hidden structure behind the distribution of integer solutions to polynomials using harmonic analysis (Fourier series) to count the solutions, which is similar to signal processing analysis, and also using Dirichlet series.

In the same way that the Riemann zeta function predicts the distribution of prime numbers, so they aim to encode polynomial solutions in a Dirichlet series, producing a means to predict the distribution of integer solutions.

Number Theory thrives through interactions with other areas of mathematics, and their work is no exception. One strategy they are currently pursuing is to combine the circle method (Hardy-Littlewood method) with descent arguments from arithmetic geometry to understand the integer solutions of equations similar to the one arising in the Taxicab number in terms of a new system of equations. In many circumstances, the new system satisfies the Local-Global Principle (Hasse Principle) and can be understood via the circle method. Others involve methods from ergodic theory and additive combinatorics.

One can even try to reinterpret such systems in terms of function field equations. 'One can hope to understand the original problem of integer solutions in terms of a function field Local-Global Principle, to be understood via a variant of the circle method in function fields', says Wooley.