Squarefree parts of polynomial values

TL;DR

This paper investigates the distribution of squarefree parts of polynomial values modulo primes, establishing conditions for divisibility, proposing a conjecture on residue classes, and proving special cases using elementary and advanced number theory.

Contribution

It introduces a conjecture about the distribution of squarefree parts of polynomial values modulo primes and proves it for degree 1 and 2 polynomials, linking higher degrees to elliptic curve conjectures.

Findings

Necessary and sufficient conditions for set S to contain multiples of p

Conjecture that S contains infinitely many representatives in each nonzero residue class for large p

Partial proofs for degrees 1, 2, and 3, and a local analogue for arbitrary degree

Abstract

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 $p$. Furthermore, 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.