Monthly Archives: December 2019

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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Solution: 2019-21 Approximate isometry

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 \( (1-\delta, 1+\delta) \).

The best solution was submitted by 고성훈 (수리과학과 2018학번). Congratulations!

Here is his solution of problem 2019-21.

A similar solution was submitted by 김태균 (수리과학과 2016학번, +3). Incomplete solutions was submitted by 박재원 (2019학번, +2), 하석민 (수리과학과 2017학번, +2).

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