Real analogue of Smale’s 17th problem

Determine whether there exists a uniform algorithm that, on average polynomial time in the input size N, computes an approximate solution on the unit sphere to a random system F(x)=(F1(x),…,Fn(x)) of n=d−1 independent real homogeneous polynomials in d variables with degrees p1,…,pn and Kostlan–Shub–Smale Gaussian coefficients as in equation (1.1), where an approximate solution means a point from which the projected Newton method converges quadratically to a real zero of F.

Background

Smale’s 17th problem asks for a uniform average-polynomial-time algorithm to find approximate zeros of n complex polynomial equations in n unknowns. In the complex setting, this has been resolved via homotopy continuation methods and subsequent derandomization.

This paper studies the real setting with random systems of independent homogeneous polynomials in the Kostlan–Shub–Smale model, focusing on solutions on the unit sphere. The authors give deterministic algorithms that succeed with high probability when n=d−O(√(d log d)) for moderate maximum degree, and for n=d−1 when the maximum degree is very large.

Despite these advances, the general real counterpart—namely, a uniform average-polynomial-time algorithm for n=d−1 in the real case—remains unresolved, as the existence of a practical algorithm analogous to the complex case is still unknown.

References

To the best of our knowledge, however, the real case of Smale's problem is completely open and this is a significant progress.

— On Smale's 17th problem over the reals  (2405.01735 - Montanari et al., 2024) in Section 1: Introduction and main result

The question of determining the characteristics of the grid (size, density, etc.) of initial conditions for Newton solving is at the heart of the famous P vs NP problem. One prototypical NP hard problem is quadratic feasibility, that is determining if a system of quadratic equations has solutions over $Q, R$ or $C$. One might attempt to solve this with Newton's method with a grid of initial conditions and wait to see if any initial point converges to a solution. Even though each instance of Newton's method takes polynomial time, the efficiency of solving depends on the number of initial points that must be considered. The P vs NP question is if a provably sufficient initial grid must have the number of initial grid points $N(n)$ exponential in $n$ the number of equations. Random choices of grid points can help in practice to keep $N(n)$ polynomially small for most systems, and de-randomization strategies are one approach to resolving P vs NP.

— Equilibria for Networks of Linear Translational Springs  (2609.03143 - Oeding et al., 2 Sep 2026) in Remark in Section 2, subsection “Newton's Method” (Section sec:newton)