Square-free values of polynomials

Abstract

It is conjectured that all separable polynomials with integers coefficients, satisfying some local conditions, take infinitely many square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article, we propose a method to show that ``cyclotomic polynomials $\Phi_{\ell}(X)$ take square free values with positive proportion". Following this method, conditionally, we do prove the Square-free conjecture for all cyclotomic polynomials. The method is applicable to some other class of polynomials as well. Moreover we show that the method readily succeeds for quadratic polynomials to give unconditional result. In the appendix, we give an elementary proof of the square-free conjecture for cyclotomic polynomials under abc conjecture.