Squarefree parts of polynomial values

Abstract

Given a separable nonconstant polynomial f pxq with integer coefficients, we consider the set S consisting of the squarefree parts of all the rational values of f pxq, 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 pxq has degree 1 or 2. If f pxq 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.

On the mod-$p$ distribution of discriminants of $G$-extensions

“…b) The special case G = C 2 in Theorem 2.1i) yields Krumm's and Pollack's result about rational points on quadratic twists on hyperelliptic curves ([16, Theorem 2]), as explained in Section 3. Compare also the earlier [15] for some even more special cases.…”

Twists of hyperelliptic curves by integers in progressions modulo $p$

“…This conjecture is proved in [4] in the case where deg f ≤ 2. Furthermore, when deg f = 3, or when deg f = 4 and f (x) has a rational root, the conjecture is shown to follow from the Parity Conjecture for elliptic curves over Q.…”

“…1 See Lemma 4.4 in [4] for details. The crucial fact we use here is that if h(x) is an irreducible factor of f (x) such that p ∤ disc h(x), then the intersection of the splitting field of h(x) and the cyclotomic field Q(ζp) is Q.…”

“…Let lc(f ) be the leading coefficient of f (x), and let g be the genus of the curve y 2 = f (x). Suppose that p is a prime satisfying (4) p ∤ lc(f ), p ∤ D, p ∤ disc f (x), and p > 4g 2 + 6g + 4.…”

“…The main object of interest in this article is the set S Q (f ) consisting of all squarefree integers d such that C d has a nontrivial rational point, i.e., an affine rational point (x 0 , y 0 ) with y 0 = 0. Specifically, we are interested in the following conjecture, which was proposed by the first author in [4].…”