Squarefree parts of polynomial values

David Krumm

doi:10.5802/jtnb.959

David Krumm ¹

¹ Department of Mathematics Colby College Waterville, ME 04901, USA

Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 3, pp. 699-724.

Abstract
Résumé

Given a separable nonconstant polynomial $f (x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f (x)$ , and study its behavior modulo primes. Fixing a prime $p$ , we determine necessary and sufficient conditions for $S$ to contain an element divisible by $p$ . We conjecture that if $p$ is large enough, then $S$ contains infinitely many representatives from every nonzero residue class modulo $p$ . The conjecture is proved by elementary means assuming $f (x)$ has degree 1 or 2. If $f (x)$ has degree 3, or if it has degree 4 and has a rational root, the conjecture is shown to follow from the parity conjecture for elliptic curves. For polynomials of arbitrary degree, a local analogue of the conjecture is proved using standard results from class field theory, and empirical evidence is given to support the global version of the conjecture.

Étant donné un polynôme non constant séparable $f (x)$ à coefficients entiers, nous considérons l’ensemble $S$ constitué des parties sans facteurs carrés de toutes les valeurs rationnelles de $f (x)$ , et étudions son comportement modulo un nombre premier. Ayant fixé un nombre premier $p$ , nous déterminons des conditions nécessaires et suffisantes pour que $S$ contienne un élément divisible par $p$ . Nous conjecturons que si $p$ est suffisamment grand, alors $S$ contient une infinité de représentants de chaque classe résiduelle non nulle modulo $p$ . Nous prouvons cette conjecture quand $f (x)$ est de degré 1 ou 2. Si $f (x)$ est de degré 3, ou s’il est de degré 4 avec une racine rationnelle, la preuve de la conjecture utilise la conjecture de parité pour les courbes elliptiques. Pour les polynômes de degré arbitraire, un analogue local de la conjecture est prouvé en utilisant des résultats standard de la théorie des corps de classe. Des résultats numériques sont aussi inclus qui confirment la version globale de la conjecture.

Received: 2014-07-17
Revised: 2015-07-15
Accepted: 2015-07-15
Published online: 2017-01-02

MR: 3610693 | Zbl: 1415.11148

DOI: 10.5802/jtnb.959

Classification: 11A07, 11R09, 11G05, 11G30
Keywords: Squarefree part, hyperelliptic curve, quadratic twist, congruent number

Author's affiliations:

David Krumm ¹

¹ Department of Mathematics Colby College Waterville, ME 04901, USA

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David Krumm. Squarefree parts of polynomial values. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 3, pp. 699-724. doi : 10.5802/jtnb.959. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.959/

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