On Smale's 17th problem over the reals

TL;DR

This paper addresses Smale's 17th problem over the real numbers, presenting a polynomial-time algorithm for solving systems of random polynomial equations with high probability under certain degree and dimension conditions.

Contribution

It introduces a polynomial-time algorithm for real polynomial systems solving, extending previous complex case results to the real setting with new probabilistic guarantees.

Findings

Efficient algorithm for n=d-1 with high probability when degrees are large.

Algorithm works for n=d - O(√d log d) when degrees are bounded by d^2.

Provides probabilistic guarantees for solutions in the real case.

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 $d^2$.