Power-free values, large deviations, and integer points on irrational curves
| dc.creator | Helfgott, H. A. | |
| dc.date | 2004-11-16 | |
| dc.date | 2007-07-03 | |
| dc.date.accessioned | 2026-07-07T08:13:38Z | |
| dc.date.available | 2026-07-07T08:13:38Z | |
| dc.description | Let $f\in \mathbb{Z}\lbrack x\rbrack$ be a polynomial of degree $d\geq 3$ 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. | |
| dc.description | 39 pages; rather major revision, with strengthened and generalized statements | |
| dc.identifier | https://arxiv.org/abs/math/0411369 | |
| dc.identifier | http://arxiv.org/abs/math/0411369 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132835 | |
| dc.subject | Number Theory | |
| dc.subject | 11N32; 11D45, 11G05, 11G30, 11N25 | |
| dc.title | Power-free values, large deviations, and integer points on irrational curves | |
| dc.type | text |