Balancing Polynomials in the Chebyshev Norm

Abstract: Given $n$ polynomials $p_1, \dots, p_n$ of degree at most $n$ with $|p_i|\infty \le 1$ for $i \in [n]$, we show there exist signs $x_1, \dots, x_n \in {-1,1}$ so that [\Big|\sum{i=1}^n x_i p_i\Big|\infty < 30\sqrt{n}, ] where $|p|\infty := \sup_{|x| \le 1} |p(x)|$. This result extends the Rudin-Shapiro sequence, which gives an upper bound of $O(\sqrt{n})$ for the Chebyshev polynomials $T_1, \dots, T_n$, and can be seen as a polynomial analogue of Spencer's "six standard deviations" theorem.