Random Sparse Polynomial Systems

TL;DR

This paper derives formulas for the expected number of roots of sparse random polynomial systems and provides bounds on the probability of large condition numbers, linking algebraic geometry, probability, and geometry.

Contribution

It introduces a new integral formula for expected roots and bounds on condition numbers for sparse random polynomial systems, connecting these to toric geometry and invariants.

Findings

Expected number of roots expressed as an integral over a mixed volume form

Bound on probability that the condition number exceeds a threshold

Extension of results to real coefficients and roots

Abstract

Let f:=(f^1,\...,f^n) be a sparse random polynomial system. This means that each f^i has fixed support (list of possibly non-zero coefficients) and each coefficient has a Gaussian probability distribution of arbitrary variance. We express the expected number of roots of f inside a region U as the integral over U of a certain mixed volume form. When U = (C^*)^n, the classical mixed volume is recovered. The main result in this paper is a bound on the probability that the condition number of f on the region U is larger than 1/epsilon. This bound depends on the integral of the mixed volume form over U, and on a certain intrinsic invariant of U as a subset of a toric manifold. Polynomials with real coefficients are also considered, and bounds for the expected number of real roots and for the condition number are given. The connection between zeros of sparse random polynomial systems, Kahler geometry, and mechanics (momentum maps) is discussed.