Polynomials are ubiquitous in mathematics, but while we learn many techniques to solve ones of low-degree in school, there's a good reason (aside from sheer tedium) as to why we don't just continue studying more and more elaborate methods for solving polynomials of higher and higher degree. For one, it turns out if there were some deterministic way to, say, find integer solutions to a general multivariate polynomial with integral coefficients, then we would essentially have an algorithm that could theoretically solve almost every problem in mathematics. Such an algorithm was proven not to exist, so we must restrict ourselves to particular settings, but for a certain class of polynomials (of arbitrarily large degree!), it turns out that know the whole story.

We begin our journey with the following question. Suppose that \(f(x)\) is a polynomial in one variable whose coefficients are integers. Consider the sequence \(f(1), f(2), \ldots, f(n), \ldots\) as \(n\) runs over the set of natural numbers. What can we say about the sequence \(\{f(n)\}\)? For example, what are the kinds of numbers that can arise? How is the sequence \(\{f(n)\}\) related to the roots of the polynomial? It turns out that the polynomials \(f(x)\) for which we can say anything at all with respect to these questions are closely intertwined to some fascinating analysis, algebra, and geometry.