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Min sum of squares of polynomial cofficients with real root

Source: Czech-Polish-Slovak 2002 Q6

April 28, 2013

algebrapolynomialinequalitiesCauchy Inequalityalgebra unsolved

Problem Statement

Let

n \ge 2

be a fixed even integer. We consider polynomials of the form

P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1

with real coefficients, having at least one real roots. Find the least possible value of

a^2_1 + a^2_2 + \cdots + a^2_{n-1}