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Generating polynomials

Source: Baltic Way 2018, Problem 5

November 6, 2018
algebrapolynomial

Problem Statement

A polynomial f(x)f(x) with real coefficients is called generating, if for each polynomial φ(x)\varphi(x) with real coefficients there exists a positive integer kk and polynomials g1(x),…,gk(x)g_1(x),\dotsc,g_k(x) with real coefficients such that φ(x)=f(g1(x))+…+f(gk(x)).\varphi(x)=f(g_1(x))+\dotsc+f(g_k(x)). Find all generating polynomials.
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