刚性延拓路径 I. 求解多项式系统的拟线性平均复杂度
Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems
AI总结：
该研究针对随机高斯多项式系统近似根的求解复杂度问题，提出基于刚性延拓路径的数值延拓算法，将平均复杂度界改进至输入规模的拟线性阶，延拓步数较此前结果实现决定性提升。
摘要：
计算随机高斯多项式系统的一个近似根，平均需要多少次运算？除了斯梅尔第17问题（询问是否存在多项式界）之外，我们证明了一个拟最优界（输入规模）^{1+o(1)}。这改进了此前已知的（输入规模）^{3/2 +o(1)}界。\n新算法依赖沿**刚性延拓路径**的数值延拓。核心思想是考虑方程的刚体运动，而非所有多项式系统构成的线性空间中的线段。这带来了更优的平均条件数，且允许更大的步长。我们证明，对于含n个方程、次数至多为D、定义在n+1个齐次变量上的随机高斯多项式系统，平均只需O(n^5 D^2)次延拓步即可算出一个近似根。相较于此前证明的平均延拓步数不优于√2^{min(n, D)}的界，这是决定性的改进。
英文摘要：
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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