论文标题
在Chebyshev Norm中平衡多项式
Balancing Polynomials in the Chebyshev Norm
论文作者
论文摘要
给定$ n $ polyenmials $ p_1，\ dots，p_n $ a $ n $，$ n $带有$ \ | p_i \ | _ \ | _ \ infty \ le 1 $ 1 $ for $ i \ in [n] $，我们显示出存在符号$ x_1，\ dots，x_1，\ dots，\ dots，x_n \ in \ in \ in \ in \ in \ { - 1,1,1,1，1,1,1，1,1,1,1,1，so so \ [\ big \ | \ sum_ {i = 1}^n x_i p_i \ big \ | _ \ _ \ infty <30 \ sqrt {n}，\]，其中$ \ | p \ | _ \ | _ \ iffty：= \ sup_ = \ sup_ {| x | x | x | x | x | x | x | \ le 1} | p（x）| $。该结果扩展了rudin-shapiro序列，该序列给出了$ O（\ sqrt {n}）$的上限，用于Chebyshev多项式$ T_1，\ dots，t_n $，可以看作是Spencer的“六个标准偏差” Theorem的多项式类似物。
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.
