On Smale's 17th problem over the reals

Andrea Montanari; Eliran Subag

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arXiv · 2405.01735

On Smale's 17th problem over the reals

Andrea Montanari·Eliran Subag

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Abstract

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 a polynomial time algorithm that finds solutions (with high probability) for $n= d -O(\sqrt{d\log d})$ if the maximal degree is bounded by $d^2$ and for $n=d-1$ if the maximal degree is larger than $d^2$.

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Andrea Montanari, Eliran Subag. 2024-05-02. On Smale's 17th problem over the reals. https://arxiv.org/abs/2405.01735

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