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

Harald A. Helfgott[1]

  • [1] Département de mathématiques et de statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 2, page 433-472
  • ISSN: 1246-7405

Abstract

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Let f ∈ ℤ [ x ] be a polynomial of degree d ≥ 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.

How to cite

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Helfgott, Harald A.. "Power-free values, large deviations, and integer points on irrational curves." Journal de Théorie des Nombres de Bordeaux 19.2 (2007): 433-472. <http://eudml.org/doc/249950>.

@article{Helfgott2007,
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.},
affiliation = {Département de mathématiques et de statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada},
author = {Helfgott, Harald A.},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {433-472},
publisher = {Université Bordeaux 1},
title = {Power-free values, large deviations, and integer points on irrational curves},
url = {http://eudml.org/doc/249950},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Helfgott, Harald A.
TI - Power-free values, large deviations, and integer points on irrational curves
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 2
SP - 433
EP - 472
AB - 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.
LA - eng
UR - http://eudml.org/doc/249950
ER -

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