Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Squarefree parts of polynomial values   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.

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The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess