On Smale's 17th problem over the reals

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 $n= d-1$,...

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Bibliographic Details
Main Authors Montanari, Andrea, Subag, Eliran
Format Journal Article
LanguageEnglish
Published 02.05.2024
Subjects
Computer Science - Data Structures and Algorithms
Computer Science - Numerical Analysis
Mathematics - Numerical Analysis
Mathematics - Optimization and Control
Mathematics - Probability
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Summary: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 $n=d-1$.
DOI:10.48550/arxiv.2405.01735