Rudin-Shapiro多项式的Mahler测度
The Mahler measure of the Rudin-Shapiro polynomials
摘要：
Littlewood多项式是每个系数都在{-1，1}中的多项式。利用Rudin-Shapiro多项式给出了在复平面的单位圆上满足显著平坦性的Littlewood多项式序列.本文证明了复平面单位圆上的Mahler测度与Rudin-Shapiro多项式的最大模具有相同的大小。证明了在复平面的单位圆的不太小的子弧上，Mahler测度与Rudin-Shapiro多项式的最大范数具有相同的大小.甚至没有非平凡的下界的马勒措施的鲁丁夏皮罗多项式已经知道。
英文摘要：
Littlewood polynomials are polynomials with each of their coefficients in {-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials have been known before.
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