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.