# Solution: 2022-10 Polynomial with root 1

Prove or disprove the following:

For any positive integer $$n$$, there exists a polynomial $$P_n$$ of degree $$n^2$$ such that

(1) all coefficients of $$P_n$$ are integers with absolute value at most $$n^2$$, and

(2) $$1$$ is a root of $$P_n =0$$ with multiplicity at least $$n$$.

The best solution was submitted by 박기찬 (KAIST 새내기과정학부 22학번, +4). Congratulations!

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For any positive integer $$n \geq 2$$, let $$B_n$$ be the group given by the following presentation$B_n = < \sigma_1, \ldots, \sigma_{n-1} | \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \sigma_i \sigma_j = \sigma_j \sigma_i >$where the first relation is for $$1 \leq i \leq n-2$$ and the second relation is for $$|i-j| \geq 2$$. Show that there exists a total order < on $$B_n$$ such that for any three elements $$a, b, c\in B_n$$, if $$a < b$$ then $$ca < cb$$.