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There exists integer n and n polynomials satisfying equality

Source: ILL 1979-38

June 2, 2011
algebrapolynomialalgebra unsolved

Problem Statement

Prove the following statement: If a polynomial f(x)f(x) with real coefficients takes only nonnegative values, then there exists a positive integer nn and polynomials g1(x),g2(x),⋯ ,gn(x)g_1(x), g_2(x),\cdots, g_n(x) such that f(x)=g1(x)2+g2(x)2+⋯+gn(x)2f(x) = g_1(x)^2 + g_2(x)^2 +\cdots+ g_n(x)^2
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