On the power-free values assumed by polynomials

International audienceIt has long been conjectured that a polynomial $f(n)$ of degree $r>1$ with integer coefficients infinitely often assumes square-free values provided that it have no non-trivial fixed square divisors and, indeed, that there is an asymptotic formula of the form $\sim A(f)x$ for the number of $n\leq x$ for which $f(n)$ is $s$th-power free. This was shown to be a consequence of the $abc$ conjecture.The present paper shows that progress is possible on the basis of conjectures relating to exponential sums, or to solutions of Pell's equations, that appear to be less far-reaching than the $abc$ conjecture. We also discuss what can be deduced from an alternative hypothesis referring only to Pell's equation, and also supply corresponding theorems relating to the case where the integer $n$ is constrained to be prime