Renormalized homotopy continuation on toric varieties is introduced as a tool for solving sparse systems of polynomial equations or sparse systems of exponential sums. The cost of continuation depends on a renormalized condition length, defined as a line integral of the condition number along all the lifted renormalized paths. The theory developed in this paper leads to a continuation algorithm tracking all the solutions between two generic systems with the same structure. The algorithm is randomized, in the sense that it follows a random path between the two systems. The probability of success is one. In order to produce an expected cost bound, several invariants depending solely on the supports of the equations are introduced. For instance, the mixed area is a quermassintegral that generalizes surface area in the same way that mixed volume generalizes ordinary volume. The facet gap measures for each 1-cone in the fan and for each support polytope, how close is the supporting hyperplane to the nearest vertex. Once the supports are fixed, the expected cost depends on the input coefficients solely through two invariants: the renormalized toric condition number and the imbalance of the absolute values of the coefficients. This leads to a nonuniform polynomial complexity bound for polynomial solving in terms of those two invariants.

中文翻译：

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稀疏多项式求解的复杂性2：重整化

介绍了复曲面变体上的重整化同伦延拓作为求解多项式方程的稀疏系统或指数和的稀疏系统的工具。继续的成本取决于重整化条件长度，定义为沿所有提升的重整化路径的条件数的线积分。本文发展的理论导致了一种连续算法跟踪具有相同结构的两个通用系统之间的所有解决方案。该算法是随机的，因为它遵循两个系统之间的随机路径。成功的概率是一。为了产生预期的成本界限，引入了几个仅取决于方程支持的不变量。例如，混合面积是一个 quermassintegral，它以与混合体积概括普通体积相同的方式概括表面积。平面间隙测量扇形中的每个 1 锥体和每个支撑多面体，支撑超平面到最近顶点的距离。一旦固定支持，预期成本仅通过两个不变量取决于输入系数：重新归一化的复曲面条件数和系数绝对值的不平衡。这导致根据这两个不变量进行多项式求解的非均匀多项式复杂性界限。预期成本仅通过两个不变量取决于输入系数：重新归一化的复曲面条件数和系数绝对值的不平衡。这导致根据这两个不变量进行多项式求解的非均匀多项式复杂性界限。预期成本仅通过两个不变量取决于输入系数：重新归一化的复曲面条件数和系数绝对值的不平衡。这导致根据这两个不变量进行多项式求解的非均匀多项式复杂性界限。