A Faster Solution to Smale's 17th Problem I: Real Binomial Systems
Grigoris Paouris, Kaitlyn Phillipson, J. Maurice Rojas
TL;DR
The paper advances Smale's 17th problem by showing that, for real Gaussian binomial systems, one can compute a real approximate root (or certify none exist) in average polynomial time with a tight arithmetic-cost bound of $O(n^2(\log n+\log\max_i d_i))$, accommodating arbitrary Gaussian variances. The approach leverages a sequence of structural reductions: a Smith normal form to diagonalize binomial systems, a monomial change of variables to transform to diagonal univariate binomials, and robust probabilistic bounds on logs of Gaussian magnitudes (via Latala-type inequalities and log-concavity estimates) to control coefficient distortions and convergence characteristics. Key technical ingredients include a fast univariate binomial solver, distortion bounds under monomial changes, and careful moment and tail analyses of sums of $\log|Z_i|$ for Gaussian $Z_i$. The results demonstrate a concrete average-case speed-up for real sparse systems in the binomial case and illuminate the obstacles to extending polynomial-time guarantees to more terms, emphasizing the role of sparsity and algebraic structure in the average-case complexity landscape.
Abstract
Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$. Recent progress on Smale's 17th Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, Bürgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of $F$ using just $N^{O(1)}$ arithmetic operations on average, where $N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$ ($=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$) is the maximum possible total number of monomial terms for such an $F$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when $F$ is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $O(n^2(\log(n)+\log\max_i d_i))$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in $n\log \max_i d_i$ when $F$ has more terms.