We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of arbitrary degrees. In the complex case and for $n= d-1$, Beltr\'{a}n and Pardo proved the existence of an efficient randomized algorithm and Lairez recently showed it can be de-randomized to produce a deterministic efficient algorithm. Here we consider the real setting, to which previously developed methods do not apply. We describe an algorithm that efficiently finds solutions (with high probability) for $n= d -O(\sqrt{d\log d})$. If the maximal degree is very large, we also give an algorithm that works up to $n=d-1$.

翻译：我们考虑在${\mathbb R}^d$中高效求解$n$个非线性方程组的问题。针对斯梅尔于1998年提出的第17问题，我们考虑一种设定，其中$n$个方程为任意次数的随机齐次多项式。在复情形且$n=d-1$时，Beltrán和Pardo证明了高效随机化算法的存在性，而Lairez最近表明该算法可以解随机化，从而产生确定性高效算法。本文考虑实数设定，此前发展的方法在此不适用。我们描述了一种算法，能够高效（以高概率）求解$n = d -O(\sqrt{d\log d})$的情形。若最高次数非常大，我们还给出了一种适用于$n=d-1$的算法。