Power-free values, large deviations, and integer points on irrational curves

Abstract

Let $f\in \mathbb{Z}[x]$ be a polynomial of degree $d\ge 3$ without roots of multiplicity $d$ or $(d-1)$. Erdős conjectured that, if $f$ satisfies the necessary local conditions, then $f(p)$ is free of $(d-1)$th powers for infinitely many primes $p$. This is proved here for all $f$ with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov's theorem from the theory of large deviations.

39 pages; rather major revision, with strengthened and generalized statements

References in corpus (1)

• On the behaviour of root numbers in families of elliptic curves