# 2019-22 Prime divisors of polynomial iterates

Let $$f = X^n + a_{n-1}X^{n-1} + \dots + a_0\in \mathbb{Z}[X]$$ be a polynomial with integer coefficients, and let $$m\in \mathbb{Z}$$.

Consider the sequence $f_0,f_1,f_2,\dots$

where $$f_0:=m$$, and $$f_i:=f(f_{i-1})$$ for all $$i\ge 1$$.

Let $$S:=\{p\in \mathbb{P}: p \text{ divides } f_i \text{ for some } i\ge 0\}$$ be the set of prime divisors of the sequence $$f_0,f_1,f_2,\dots$$.

Assume that $$S$$ is finite, but $$\{f_i\mid i\ge 0\}$$ is infinite. Show that $$f=X^n$$.

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Let $$A$$ be an $$m \times n$$ matrix and $$\delta \in (0, 1)$$. Suppose that $$\| A^T A – I \| \leq \delta$$. Prove that all singular values of $$A$$ are contained in the interval $$(1-\delta, 1+\delta)$$.