We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F \in L[Y]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for $F$ and algorithmic and bit-complexity results for the problem of isolating its roots. For the latter problem we follow a common approach based on univariate root isolation algorithms. For the particular case where $F$ does not have multiple roots, we achieve a bit-complexity in $\tilde{\mathcal{O}}_B(n d^{2n+2}(d+n\tau))$, where $d$ is the total degree and $\tau$ is the bitsize of the involved polynomials.In the general case we need to enhance our algorithm with a preprocessing step that determines the number of distinct roots of $F$. We follow a numerical, yet certified, approach that has bit-complexity $\tilde{\mathcal{O}}_B(n^2d^{3n+3}\tau + n^3 d^{2n+4}\tau)$.

翻译：本文研究当多项式系数位于多域扩张中时的单变量根隔离问题。我们考虑多项式 $F \in L[Y]$，其中 $L$ 是 $\mathbb{Q}$ 的多重代数扩张。我们给出了 $F$ 的聚合界，以及针对其根隔离问题的算法与比特复杂度结果。对于后者，我们采用基于单变量根隔离算法的通用方法。在 $F$ 无重根的特定情形下，我们实现了 $\tilde{\mathcal{O}}_B(n d^{2n+2}(d+n\tau))$ 的比特复杂度，其中 $d$ 为总次数，$\tau$ 为所涉及多项式的比特大小。在一般情形中，我们需通过预处理步骤增强算法，以确定 $F$ 的不同根的数量。我们采用数值化但可验证的方法，其比特复杂度为 $\tilde{\mathcal{O}}_B(n^2d^{3n+3}\tau + n^3 d^{2n+4}\tau)$。