On Square-Free Values of Polynomials over Integers and Function Fields

Abstract

Square-free values are of significant interest to number theorists, partly due to their close connection to the Mobius function. However, little is known about square-free values of polynomials with integer coefficient when the degree of the polynomial is greater than 3. For instance, it has yet to be proved that the polynomial f(x) = x^4 + 2 takes infinitely many square-free values, let alone that an asymptotic density of square- free values is known. The main purpose of this thesis is to show how to prove the density of square-free values for polynomials of degree 1 and 2, after which the focus shifts to polynomials in one variable over the field F_q. We show that, when considering the latter, a certain asymptotic density can be proved for square-free polynomials of any degree. Furthermore, we compare this proof to the strategy used to prove the asymptotic density of polynomials with integer coefficients. Finally, we investigate the density of square-free values of polynomials with large varying coefficients, making use of Brun’s sieve, and again find similar densities.