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2022 Alg/NT Div 1 P6

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February 28, 2022
algebranumber theory

Problem Statement

Find the probability such that when a polynomial in Z2027[x]\mathbb Z_{2027}[x] having degree at most 20262026 is chosen uniformly at random, x2027−x∣Pk(x)−x  ⟺  2021∣kx^{2027}-x | P^{k}(x) - x \iff 2021 | k (note that 20272027 is prime). Here Pk(x)P^k(x) denotes PP composed with itself kk times.
Proposed by Grant Yu
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