拟随机多线性多项式
Quasi-random multilinear polynomials
摘要：
我们考虑多线性Littlewood多项式，多项式的$n$变量，其中一个指定的一组单式$U$有$\pm 1$系数，和所有其他系数是$0$。我们提供上界和下界（这是接近的程度低于$\log n$ $U$）的最小值，在多项式$h$一致$U$，最大值的$|h|$超过$\pm 1$分配给变量。(This是一个变种的问题所提出的埃尔德什关于最大的单位磁盘上的一元多项式的程度与单位系数。我们概述了准随机图和超图的理论和统计力学模型的连接。我们的方法依赖于Gale-Berlekamp游戏的分析;在建设性的通用链接方法的一面;在一个欣钦型不等式的多项式的程度大于$1$ ;和伯恩斯坦的近似理论不等式。
英文摘要：
We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$ of degree below $\log n$) on the minimum, over polynomials $h$ consistent with $U$, of the maximum of $|h|$ over $\pm 1$ assignments to the variables. (This is a variant of a question posed by Erdös regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale-Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than $1$; and on Bernstein's approximation theory inequality.
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