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Existence of polynomial satisfying conditions

Source: China TSTST 3 Day 2 Problem 1

March 17, 2017
algebrapolynomial

Problem Statement

Show that there exists a degree 5858 monic polynomial P(x)=x58+a1x57+⋯+a58P(x) = x^{58} + a_1x^{57} + \cdots + a_{58} such that P(x)P(x) has exactly 2929 positive real roots and 2929 negative real roots and that log⁡2017∣ai∣\log_{2017} |a_i| is a positive integer for all 1≤i≤581 \leq i \leq 58.
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