由特征标导出的多项式的渐近$L^4$范数
Asymptotic $L^4$ norm of polynomials derived from characters
摘要：
Littlewood研究了系数在$\{-1,1\}$中的多项式（Littlewood多项式），看看它们在单位圆上的范数比$||f||_4/||f||_2$可以变得多小，如$deg(f)\to\infty$。一个小的限制相当于多项式系数的相关二进制序列的均方自相关的缓慢增长。阵列和高维对象的自相关问题也被研究;它是多变量多项式的自然推广。在这里，我们发现，对于每个$n > 1$，一个家庭的$n$变量Littlewood多项式与较低的渐近$||f||_4/||f||_2$比任何已知的迄今为止。我们发现这些通过广泛的调查，不可行的与以前的方法，多项式的系数来自有限域字符。这是第一次，最低已知的渐近比规范$||f||_4/||f||_2$多变量多项式$f(z_1,...,z_n)$是严格小于可以通过使用产品$f_1(z_1)... f_n(z_n)$的最知名的一元多项式。
英文摘要：
Littlewood investigated polynomials with coefficients in $\{-1,1\}$ (Littlewood polynomials), to see how small their ratio of norms $||f||_4/||f||_2$ on the unit circle can become as $deg(f)\to\infty$. A small limit is equivalent to slow growth in the mean square autocorrelation of the associated binary sequences of coefficients of the polynomials. The autocorrelation problem for arrays and higher dimensional objects has also been studied; it is the natural generalization to multivariable polynomials. Here we find, for each $n > 1$, a family of $n$-variable Littlewood polynomials with lower asymptotic $||f||_4/||f||_2$ than any known hitherto. We discover these through a wide survey, infeasible with previous methods, of polynomials whose coefficients come from finite field characters. This is the first time that the lowest known asymptotic ratio of norms $||f||_4/||f||_2$ for multivariable polynomials $f(z_1,...,z_n)$ is strictly less than what could be obtained by using products $f_1(z_1)... f_n(z_n)$ of the best known univariate polynomials.
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