How many operations do we need on the average to compute an approximate rootof a random Gaussian polynomial system? Beyond Smale's 17th problem that askedwhether a polynomial bound is possible, we prove a quasi-optimal bound$ext{(input size)}^{1+o(1)}$. This improves upon the previously known$ext{(input size)}^{rac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along emph{rigidcontinuation paths}. The central idea is to consider rigid motions of theequations rather than line segments in the linear space of all polynomialsystems. This leads to a better average condition number and allows for biggersteps. We show that on the average, we can compute one approximate root of arandom 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 adecisive improvement over previous bounds that prove no better than$sqrt{2}^{min(n, D)}$ continuation steps on the average.
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机译：我们需要多少操作，以计算一个随机高斯多项式系统的近似rootiof？超越Smale的第17个问题，询问多项式绑定的，我们证明了一个准优选的$ text {（输入大小）} ^ {1 + o（1）} $。这改善了先前已知的$ text {（输入大小）} ^ { frac32 + O（1）} $绑定。新算法依赖于 EMPH {刚性发源道}的数值延续。中心思想是考虑所有多项式系统的线性空间中的刚性的刚性运动而不是线段。这导致更好的平均条件号，允许BigGerseps。我们展示了平均值，我们可以计算〜$ N $等式的Arandom高斯多项式系统的一个近似根，以$ n + 1 $ ocogenceblbles以$ o（n ^ 5 d ^ 2） $延续步骤。这是对之前的界限的adeciSive改进，该界限不得比$ sqrt {2} ^ { min（n，d）}在平均值上的连续性步骤。
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