乘积多项式和因子多项式的范数
Norms of products and factors polynomials
摘要：
研究多项式的一致范数的乘积与其乘积的范数之间的不等式。推广了单位圆盘上的Gel'fond-Mahler不等式和线段$[-1,1]$上的Kneser-Borwein不等式，证明了平面上任意紧集上代数多项式乘积的模的一个渐近尖锐不等式.应用类似的技巧，我们得到一个相关的不等式的范数的一个单一的一元因子的首一多项式。用位势理论方法得到了这两个不等式的最佳常数。我们还考虑应用的一般结果的情况下，一个磁盘和一个部分。
英文摘要：
We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. Generalizing Gel'fond-Mahler inequality for the unit disk and Kneser-Borwein inequality for the segment $[-1,1]$, we prove an asymptotically sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set in plane. Applying similar techniques, we produce a related inequality for the norm of a single monic factor of a monic polynomial. The best constants in both inequalities are obtained by potential theoretic methods. We also consider applications of the general results to the cases of a disk and a segment.
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