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Numerical Nonlinear Algebra

Daniel J. Bates, Paul Breiding, Tianran Chen, Jonathan D. Hauenstein, Anton Leykin, Frank Sottile

TL;DR

The paper surveys numerical nonlinear algebra, detailing polynomial and polyhedral homotopy continuation as core computational tools for solving polynomial systems and representing algebraic varieties. It emphasizes numerical algebraic geometry concepts like witness sets, numerical irreducible decomposition, regeneration, and monodromy, along with rigorous certification via Smale’s alpha-theory and Krawczyk’s interval methods. Bernstein/BKK bounds and mixed volumes underpin polyhedral approaches that exploit sparsity to reduce path-tracking effort. The text showcases applications in synchronization (Kuramoto), enumerative geometry, and computer vision, illustrating how these numerical methods enable practical solution counting, certification, and robust minimal-problem solving in real-world settings.

Abstract

Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.

Numerical Nonlinear Algebra

TL;DR

The paper surveys numerical nonlinear algebra, detailing polynomial and polyhedral homotopy continuation as core computational tools for solving polynomial systems and representing algebraic varieties. It emphasizes numerical algebraic geometry concepts like witness sets, numerical irreducible decomposition, regeneration, and monodromy, along with rigorous certification via Smale’s alpha-theory and Krawczyk’s interval methods. Bernstein/BKK bounds and mixed volumes underpin polyhedral approaches that exploit sparsity to reduce path-tracking effort. The text showcases applications in synchronization (Kuramoto), enumerative geometry, and computer vision, illustrating how these numerical methods enable practical solution counting, certification, and robust minimal-problem solving in real-world settings.

Abstract

Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.
Paper Structure (31 sections, 6 theorems, 68 equations, 12 figures, 1 table, 3 algorithms)

This paper contains 31 sections, 6 theorems, 68 equations, 12 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. What is polynomial homotopy continuation?
  3. The solution to a system of polynomial equations
  4. The Parameter Continuation Theorem
  5. The total degree homotopy
  6. Path-tracking
  7. Squaring up
  8. Polyhedral homotopy
  9. Bernstein's bound
  10. Polyhedral homotopy of Huber and Sturmfels
  11. Mixed cells
  12. The Polyhedral Homotopy Algorithm
  13. Solving binomial systems
  14. Numerical algebraic geometry
  15. More on linear slices
  16. ...and 16 more sections

Key Result

Theorem 1

With these definitions and when $d=0$, suppose that $\gamma(t)\colon [0, 1] \rightarrow \mathbb{C}^k$ is a continuous path. At points $t\in[0,1]$ with $\gamma(t)\in U$ where $\gamma$ is differentiable, $\mathbf{x}(t)$ is differentiable.

Figures (12)

  • Figure 1: Homotopy Paths.
  • Figure 2: Euler prediction followed by Newton corrections. The image is adapted from BT_Intro (we thank Sascha Timme for allowing us to use his figure).
  • Figure 3: The zero sets of $f_1$ (blue) and $f_2$ (red) together with their intersections
  • Figure 4: Two views of the lower hull of the lift and the induced mixed subdivision with mixed cells labeled by corresponding cocharacters.
  • Figure 5: A reducible variety, defined implicitly by (\ref{['ex:nag']}).
  • ...and 7 more figures

Theorems & Definitions (32)

  • Theorem 1: Parameter Continuation Theorem
  • Example 2
  • Example 3
  • Remark 4
  • Example 5
  • Remark 6
  • Example 7
  • Theorem 8: Bernstein
  • Example 9
  • Remark 10
  • ...and 22 more