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Random Sparse Polynomial Systems

Gregorio Malajovich, J. Maurice Rojas

TL;DR

The paper studies zeros of random sparse polynomial systems with fixed supports and Gaussian coefficients, showing that the expected root count in a region $U$ is $\frac{n!}{\pi^n} \int_U d\mathcal{T}^n$ and connecting this to mixed volumes via a toric/Kähler framework. It builds a phase space on a toric manifold with a momentum map $g_A$ and Veronese embedding, enabling a Bernstein-type analysis of root counts and a precise bound on the condition number expressed as an inverse distance to the discriminant along evaluation fibers. The authors extend the results to real polynomials, providing bounds for the expected number of real roots and the real condition number, and discuss unmixed versus mixed cases, including dense and multi-homogeneous scenarios. The work bridges sparse polynomial root theory with symplectic and Kähler geometry, offering a probabilistic generalization of Bernstein’s theorem and informing the stability and complexity of sparse homotopy methods.

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.

Random Sparse Polynomial Systems

TL;DR

The paper studies zeros of random sparse polynomial systems with fixed supports and Gaussian coefficients, showing that the expected root count in a region is and connecting this to mixed volumes via a toric/Kähler framework. It builds a phase space on a toric manifold with a momentum map and Veronese embedding, enabling a Bernstein-type analysis of root counts and a precise bound on the condition number expressed as an inverse distance to the discriminant along evaluation fibers. The authors extend the results to real polynomials, providing bounds for the expected number of real roots and the real condition number, and discuss unmixed versus mixed cases, including dense and multi-homogeneous scenarios. The work bridges sparse polynomial root theory with symplectic and Kähler geometry, offering a probabilistic generalization of Bernstein’s theorem and informing the stability and complexity of sparse homotopy methods.

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.

Paper Structure

This paper contains 20 sections, 26 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Expected Number of Roots
  3. The Condition Number
  4. Real Polynomials
  5. Acknowledgements
  6. Symplectic Geometry and Polynomial Systems
  7. About Symplectic Geometry
  8. Toric Actions and the Momentum Map
  9. More Properties of the Momentum Map
  10. Evaluation Map and Condition Matrix
  11. The Condition Number
  12. Proof of Theorem \ref{['condnumber']}
  13. Idea of the Proof of Theorem \ref{['mixed-cond']}
  14. From Gaussians to Multiprojective Spaces
  15. Real Polynomials
  16. ...and 5 more sections

Key Result

Theorem 1

Let $A_1, \cdots, A_n$ and $C_1, \cdots, C_n$ be as above. For each $i=1,\cdots,n$, let $f_i$ be an (independently distributed) normal random polynomial in $(\mathcal{F}_{A_i}, \langle\cdot,\cdot\rangle_{ {C_i^{-1}}})$. Let $U$ be a measurable region of $\mathcal{T}^{n}$. Then, the expected number o

Figures (1)

  • Figure 1: Geometric interpretation of $Dv_A^{\alpha}$

Theorems & Definitions (62)

  • Theorem 1
  • Example 1
  • Theorem 2: Condition number theorem
  • Corollary 2.1: Condition number theorem for unmixed systems
  • Corollary 2.2: Condition number theorem for dense systems
  • Definition 1
  • Theorem 3: Expected value of the condition number
  • Corollary 3.1
  • Remark 1
  • Theorem 4
  • ...and 52 more