High Probability Analysis of the Condition Number of Sparse Polynomial Systems
Gregorio Malajovich, J. Maurice Rojas
TL;DR
The paper addresses the problem of bounding the probability that the condition number $\boldsymbol{\mu}(f,\zeta)$ of a random sparse polynomial system $f$ with fixed supports becomes large in a toric region $U$, by relating conditioning to the geometry of toric Kähler manifolds. It develops a unified framework using toric forms $\omega_{A_i}$, a mixed metric $d_{\mathbb P}$, and a mixed dilation $\kappa_U$ to bound conditioning probabilities and to count roots via wedge-integrals that recover Bernstein–Kushniren-type results; in particular, the average number of complex roots in $U$ is $\frac{(-1)^{n(n-1)/2}}{\pi^n}\int_U \bigwedge_i \omega_{A_i}$, with a simple mixed-volume interpretation when $U=(\mathbb C^*)^n$. The results extend classical dense-case bounds (Shub–Smale/Kostlan) to sparse, toric settings and yield real-root bounds that scale with the region size and the mixed-volume geometry. Overall, the work provides explicit probabilistic guarantees and a geometric bridge between conditioning, root counts, and toric/mixed-volume theory, with implications for the complexity analysis of sparse homotopy methods.
Abstract
Let F:=(f_1,...,f_n) be a random polynomial system with fixed n-tuple of supports. Our main result is an upper bound on the probability that the condition number of f in a region U is larger than 1/epsilon. The bound depends on an integral of a differential form on a toric manifold and admits a simple explicit upper bound when the Newton polytopes (and underlying covariances) are all identical. We also consider polynomials with real coefficients and give bounds for the expected number of real roots and (restricted) condition number. Using a Kahler geometric framework throughout, we also express the expected number of roots of f inside a region U as the integral over U of a certain {\bf mixed volume} form, thus recovering the classical mixed volume when U = (C^*)^n.