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What if FE met Z[x]?

Source: 2023 IRN-SGP-TWN Friendly Math Competition P6

July 16, 2023
functional equationpolynomialpolynomial with integer coeffialgebranumber theory

Problem Statement

Z[x]\mathbb{Z}[x] represents the set of all polynomials with integer coefficients. Find all functions f:Z[x]→Z[x]f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x] such that for any 2 polynomials P,QP,Q with integer coefficients and integer rr, the following statement is true. P(r)∣Q(r)  ⟺  f(P)(r)∣f(Q)(r).P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).
(We define a∣ba|b if and only if b=zab=za for some integer zz. In particular, 0∣00|0.)
Proposed by the4seasons.
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