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).

中文翻译：

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代数几何中的函数范数，条件数和数值算法

在数值线性代数中，一种公认的做法是选择一个利用当前问题结构的准则，以优化准确性或计算复杂性。在数值多项式代数中，单个规范（归因于Weyl）主导着文献。本文着重将$ L_p $范数用于数值代数几何，重点是$ L _ {\ infty} $。这个经典的思想在分析众多迭代算法执行的步骤数方面产生了重大改进。特别是，我们展示了三种算法，尽管计算$ L _ {\ infty} $-范数很复杂，但是使用$ L_p $ -norms却大大降低了计算复杂度：实数代数几何中基于细分的算法，用于计算同源性半代数集