How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along \emph{rigid continuation paths}. The central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems. This leads to a better average condition number and allows for bigger steps. We show that on the average, we can compute one approximate root of a random Gaussian polynomial system of~$n$ equations of degree at most $D$ in $n+1$ homogeneous variables with $O(n^5 D^2)$ continuation steps. This is a decisive improvement over previous bounds that prove no better than $\sqrt{2}^{\min(n, D)}$ continuation steps on the average.

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刚性连续路径 I. 求解多项式系统的拟线性平均复杂度

我们平均需要多少次运算来计算随机高斯多项式系统的近似根？除了询问多项式边界是否可能的 Smale 第 17 个问题之外，我们证明了一个准最优边界 $\text{(input size)}^{1+o(1)}$。这改进了先前已知的 $\text{(input size)}^{\frac32 +o(1)}$ 边界。新算法依赖于沿 \emph {刚性延续路径}的数值延续。中心思想是考虑方程的刚性运动，而不是所有多项式系统的线性空间中的线段。这会导致更好的平均条件数并允许更大的步骤。我们表明，平均而言，我们可以用 $O(n^5 D^2)$ 的连续步骤计算一个随机高斯多项式系统的一个近似根，该系统包含在 $n+1$ 个齐次变量中最多 $D$ 的阶数方程。这是对先前边界的决定性改进，证明不比 $\sqrt{2}^{\min(n, D)}$ 平均连续步骤更好。