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p(|z|) <= x^4 + y^4 where z=x + iy , p(0) = 0

Source: Spanish Mathematical Olympiad 1981 P4

December 23, 2022
complex numberspolynomialalgebrainequalities

Problem Statement

Determine a polynomial of non-negative real coefficients that satisfies the following two conditions: p(0)=0,p(∣z∣)≤x4+y4,p(0) = 0, p(|z|) \le x^4 + y^4, being ∣z∣|z| the module of the complex number z=x+iyz = x + iy .
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