论文标题
关于多项式条件数量的最小值
On the minimum value of the condition number of polynomials
论文作者
论文摘要
在1993年，Shub和Smale提出了一个问题，即找到一系列单变量的多项式$ n $，条件号与上述为$ n $。在C.beltán，U。Etayo，J。Marzo和J. Ortega-cerdà的先前论文中，事实证明，条件数的最佳价值是$ o（\ sqrt {n}）$的形式，Shub和Smale所需的顺序是由封闭式搜索（对于足够的$ n \ geqslant n __0 $）的封闭式搜索，并且其余案件。在本文中，我们找到了隐藏在$ o（\ sqrt {n}）$项中的常数的具体估计，我们描述了一系列多项式的简单公式，其条件编号最多为$ n $，对所有$ n = 4m^2 $有效，$ n = 4m^2 $，$ m $ m $ a $ a a个正integer。
In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Beltán, U. Etayo, J. Marzo and J. Ortega-Cerdà, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the sequence demanded by Shub and Smale was described by a closed formula (for large enough $N\geqslant N_0$ with $N_0$ unknown) and by a search algorithm for the rest of the cases. In this paper we find concrete estimates for the constant hidden in the $O(\sqrt{N})$ term and we describe a simple formula for a sequence of polynomials whose condition number is at most $N$, valid for all $N=4M^2$, with $M$ a positive integer.
