稀疏多项式求解的复杂性：在紧致化托里奇流形上的同调及条件度量
Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric
AI总结：
本文研究了通过同调持续法求解稀疏多项式方程组的成本。首先，通过n个单项基确定n元多项式方程组的空间。这些系统的根的自然定位是某种托里奇流形。该流形是依赖于单项基的$(\mathbb C\setminus\{0\})^n$的紧致化。在该托里奇流形上定义了托里奇牛顿算子。Smale的alpha理论被推广以提供二次收敛的标准。定义了两个条件数并在此设定中获得了高阶导数估计。牛顿算子和相关的条件数在与动量映射相关的群作用下保持不变。给出了一个同调算法，并证明其在提升的同调路径的条件长度上线性终止。这扩展了Shub（2009）的结果。
摘要：
本文研究了通过同调持续法求解稀疏多项式方程组的成本。首先，通过n个单项基确定n元多项式方程组的空间。这些系统的根的自然定位是某种托里奇流形。该流形是依赖于单项基的$(\mathbb C\setminus\{0\})^n$的紧致化。在该托里奇流形上定义了托里奇牛顿算子。Smale的alpha理论被推广以提供二次收敛的标准。定义了两个条件数并在此设定中获得了高阶导数估计。牛顿算子和相关的条件数在与动量映射相关的群作用下保持不变。给出了一个同调算法，并证明其在提升的同调路径的条件长度上线性终止。这扩展了Shub（2009）的结果。
英文摘要：
This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the roots of those systems is known to be a certain toric variety. This variety is a compactification of $(\mathbb C\setminus\{0\})^n$, dependent on the monomial bases. A toric Newton operator is defined on that toric variety. Smale's alpha theory is generalized to provide criteria of quadratic convergence. Two condition numbers are defined and a higher derivative estimate is obtained in this setting. The Newton operator and related condition numbers turn out to be invariant through a group action related to the momentum map. A homotopy algorithm is given, and is proved to terminate after a number of Newton steps which is linear on the condition length of the lifted homotopy path. This generalizes a result from Shub (2009).
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