论文标题
A $ p $ - 亚法笛卡尔求解器：Strassman求解器
A $p$-adic Descartes solver: the Strassman solver
论文作者
论文摘要
解决多项式是数学中的基本计算问题。在实际环境中，我们可以使用笛卡尔的符号规则有效地隔离无方形的真实多项式的真实根。在本文中，我们将此方法转化为$ p $ -ADIC世界。我们展示了descarts符号规则的$ p $ - adic类似物，Strassman的定理是如何导致算法来隔离无方形$ p $ p $ adiC多项式的根源的算法。此外，我们表明该算法以$ \ Mathcal {o}（d^2 \ log^3d）$运行 - 随机$ p $ -p $ - adic多项式的时间$ d $。为了执行此分析，我们将基于条件的复杂性框架从实际/复杂数值代数几何形状引入$ p $ -Adic数值代数几何形状。
Solving polynomials is a fundamental computational problem in mathematics. In the real setting, we can use Descartes' rule of signs to efficiently isolate the real roots of a square-free real polynomial. In this paper, we translate this method into the $p$-adic worlds. We show how the $p$-adic analog of Descartes' rule of signs, Strassman's theorem, leads to an algorithm to isolate the roots of a square-free $p$-adic polynomial. Moreover, we show that this algorithm runs in $\mathcal{O}(d^2\log^3d)$-time for a random $p$-adic polynomial of degree $d$. To perform this analysis, we introduce the condition-based complexity framework from real/complex numerical algebraic geometry into $p$-adic numerical algebraic geometry.
