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Sum of absoulte values of polynomial at roots of the other

Source: KoMaL A. 867

January 13, 2024
algebrapolynomial

Problem Statement

Let p(x)p(x) be a monic integer polynomial of degree nn that has nn real roots, α1,α2,…,αn\alpha_1,\alpha_2,\ldots, \alpha_n. Let q(x)q(x) be an arbitrary integer polynomial that is relatively prime to polynomial p(x)p(x). Prove that ∑i=1n∣q(αi)∣≥n.\sum_{i=1}^n \left|q(\alpha_i)\right|\ge n.
Submitted by Dávid Matolcsi, Berkeley
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