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Integer roots preserved under linear function of polynomial

Source: USEMO 2019/2

May 23, 2020
functionalgebrapolynomialUSEMO

Problem Statement

Let Z[x]\mathbb{Z}[x] denote the set of single-variable polynomials in xx with integer coefficients. Find all functions θ:Z[x]→Z[x]\theta : \mathbb{Z}[x] \to \mathbb{Z}[x] (i.e. functions taking polynomials to polynomials) such that
[*] for any polynomials p,q∈Z[x]p, q \in \mathbb{Z}[x], θ(p+q)=θ(p)+θ(q)\theta(p + q) = \theta(p) + \theta(q); [*] for any polynomial p∈Z[x]p \in \mathbb{Z}[x], pp has an integer root if and only if θ(p)\theta(p) does.
Carl Schildkraut
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