We present a new data structure to approximate accurately and efficiently a polynomial $f$ of degree $d$ given as a list of coefficients. Its properties allow us to improve the state-of-the-art bounds on the bit complexity for the problems of root isolation and approximate multipoint evaluation. This data structure also leads to a new geometric criterion to detect ill-conditioned polynomials, implying notably that the standard condition number of the zeros of a polynomial is at least exponential in the number of roots of modulus less than $1/2$ or greater than $2$.Given a polynomial $f$ of degree $d$ with $\|f\|_1 \leq 2^\tau$ for $\tau \geq 1$, isolating all its complex roots or evaluating it at $d$ points can be done with a quasi-linear number of arithmetic operations. However, considering the bit complexity, the state-of-the-art algorithms require at least $d^{3/2}$ bit operations even for well-conditioned polynomials and when the accuracy required is low. Given a positive integer $m$, we can compute our new data structure and evaluate $f$ at $d$ points in the unit disk with an absolute error less than $2^{-m}$ in $\widetilde O(d(\tau+m))$ bit operations, where $\widetilde O(\cdot)$ means that we omit logarithmic factors. We also show that if $\kappa$ is the absolute condition number of the zeros of $f$, then we can isolate all the roots of $f$ in $\widetilde O(d(\tau + \log \kappa))$ bit operations. Moreover, our algorithms are simple to implement. For approximating the complex roots of a polynomial, we implemented a small prototype in \verb|Python/NumPy| that is an order of magnitude faster than the state-of-the-art solver \verb/MPSolve/ for high degree polynomials with random coefficients.

中文翻译：

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用于单变量多项式逼近的新数据结构以及在根隔离、数值多点评估和其他问题中的应用

我们提出了一种新的数据结构，以准确有效地近似以系数列表形式给出的多项式 $f$，阶数为 $d$。它的特性使我们能够针对根隔离和近似多点评估问题改进位复杂度的最新界限。这种数据结构还导致了一种新的几何标准来检测病态多项式，这意味着多项式零点的标准条件数至少是小于 $1/2$ 或大于 $1/2$ 的模根数的指数$2$. 给定一个多项式 $f$ 的次数为 $d$ 和 $\|f\|_1 \leq 2^\tau$ for $\tau \geq 1$，隔离其所有复根或在 $d$ 处求值点可以通过准线性数量的算术运算来完成。但是，考虑到位复杂度，最先进的算法需要至少 $d^{3/2}$ 位运算，即使对于条件良好的多项式并且在所需的精度较低时也是如此。给定一个正整数 $m$，我们可以计算我们的新数据结构，并在单位磁盘中的 $d$ 个点评估 $f$，绝对误差小于 $2^{-m}$ in $\widetilde O(d( \tau+m))$ 位运算，其中 $\widetilde O(\cdot)$ 意味着我们省略了对数因子。我们还表明，如果 $\kappa$ 是 $f$ 的零点的绝对条件数，那么我们可以在 $\widetilde O(d(\tau + \log \kappa)) 中隔离 $f$ 的所有根$位操作。此外，我们的算法很容易实现。为了逼近多项式的复根，我们在 \verb|Python/NumPy| 中实现了一个小原型 对于具有随机系数的高次多项式，这比最先进的求解器 \verb/MPSolve/ 快一个数量级。