Balancing Polynomials in the Chebyshev Norm
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Given n polynomials p_1, …, p_n of degree at most n with p_i_∞≤ 1 for i ∈ [n], we show there exist signs x_1, …, x_n ∈{-1,1} so that ∑_i=1^n x_i p_i_∞ < 30√(n), where p_∞ := sup_|x| ≤ 1 |p(x)|. This result extends the Rudin-Shapiro sequence, which gives an upper bound of O(√(n)) for the Chebyshev polynomials T_1, …, T_n, and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.
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