In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand in order to optimize accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty}$. This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty}$-norm, the use of $L_p$-norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry, and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale's 17th problem).

翻译：在数字线性代数中,一种既定做法是选择一种规范,利用手头问题的结构,以优化准确性或计算复杂性。在数字多数值代数中,一个单一规范(归因于Weyl)主导文献。本文章开始使用$L_p$规范用于数值代数几何,重点是$L ⁇ infty}美元。这一古典思想在分析许多迭代运算法所执行的步骤数量方面带来很大的改进。特别是,我们展示了三种算法,尽管计算$L ⁇ infty}-norm十分复杂,但使用$L_p$-norms却大大降低了计算复杂性:在计算半数数组的同性时,一种基于实际代数的亚视算法,一种在计算数数组中广为人知的计量算法,以及计算复杂二次多式系统零数(Smaly's第17个问题的一个特别案例)的计算。