Repulsive action in Number theory

Mathematics is a tool to conquer Physics and nobody can deny that fact. The real scenario is that Physics develops situations where we have to knock on the door of mathematics in order to find a truly satisfying solution. But in this case, Mathematics is developing some understanding which leads to something physical in meaning. Mathematicians are crazy, no doubt! They just want to analyze the situation no matter how crazy that sounds. This goes with an idea of higher dimensions, we still don't know anything about their existence but they have been developing its mathematics for a very long time. But here we are not interested in a Higher dimension or topology. Those topics are way too humongous to deal with. We are interested in Number theory, Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." 

SchinzelZassenhaus conjecture in Number theory is all about the geometry of values of polynomials. It's more like graphing certain values, they will be spread out in an exact way, looking as if they are pushing each other apart. The roots of a polynomial function relate to each other. When they are graphed then the roots fall on the vertices of a regular polygon. There exist something called 'Cyclotomic polynomials', these are a special family of polynomials that cannot be factored into smaller polynomials. Now if we plot a polynomial function of some degree 'n' and make a circle with a center at the origin then we will find out that there exist roots of the function (equation when equates to zero) on the circumference of the circle. The roots are definitely located inside and outside of the circle. But there is something peculiar about this picture. This more or less depicts the fact that the numbers (roots) lying outside the circle are experiencing a repulsive nature from the points inside the circle. 

                               

                                 

Source: Quanta Magazine

Now there is no doubt about the repulsion but the entire discussion at present is all about the dimension of the circle. The size of the circle matters a lot about the roots which are inside and outside the circle. At first, it was thought that the radius of the circle is inversely proportional to the degree of the polynomial. But that idea had some flaws so it was corrected to a factor of log2 and some power of the polynomial. This is one pretty theory that bridges and depicts two beautiful concepts of natural sciences.