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Polynomial Homotopies for Dense, Sparse and Determinantal Systems

Jan Verschelde

TL;DR

The paper develops optimal polynomial homotopy continuation frameworks for three polynomial-system classes—dense, sparse, and determinantal—by constructing start systems whose regular solutions match the intrinsic root counts and then deforming to target systems via well-behaved homotopies. It unifies algebraic counts (Bézout, Bernstein–Kushnirenko–Khovanskii, and Schubert-type counts) with geometric deformations (product, toric, Pieri) and polyhedral techniques (Cayley trick, mixed subdivisions, endgames) to achieve efficient, robust numerical solution. The practical impact includes a detailed pipeline (from start-system construction to endgames) and a software framework (PHC) augmented by a database of real-world problems, enabling scalable solving and robust verification. The work highlights the interplay between algebraic geometry and numerical continuation, advancing both theory and computational tools for solving large-scale polynomial systems in science and engineering, with ongoing challenges in real solutions and meta-approaches to counting and deformation.

Abstract

Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and applications are discussed.

Polynomial Homotopies for Dense, Sparse and Determinantal Systems

TL;DR

The paper develops optimal polynomial homotopy continuation frameworks for three polynomial-system classes—dense, sparse, and determinantal—by constructing start systems whose regular solutions match the intrinsic root counts and then deforming to target systems via well-behaved homotopies. It unifies algebraic counts (Bézout, Bernstein–Kushnirenko–Khovanskii, and Schubert-type counts) with geometric deformations (product, toric, Pieri) and polyhedral techniques (Cayley trick, mixed subdivisions, endgames) to achieve efficient, robust numerical solution. The practical impact includes a detailed pipeline (from start-system construction to endgames) and a software framework (PHC) augmented by a database of real-world problems, enabling scalable solving and robust verification. The work highlights the interplay between algebraic geometry and numerical continuation, advancing both theory and computational tools for solving large-scale polynomial systems in science and engineering, with ongoing challenges in real solutions and meta-approaches to counting and deformation.

Abstract

Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and applications are discussed.

Paper Structure

This paper contains 12 sections, 10 theorems, 31 equations, 11 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Three Classes of Polynomial Systems
  3. The Principles of Polynomial Homotopy Continuation Methods
  4. The Geometry of the Deformations
  5. Root Counts and Start Systems
  6. Dense Polynomials modeled by Highest Degrees
  7. Mixed Subdivisions of Newton Polytopes to compute Mixed Volumes
  8. Sparse Polynomial Systems solved by Polyhedral Homotopies
  9. Determinantal Polynomials arising in Enumerative Geometry
  10. Numerical Software for Solving Polynomial Systems
  11. The Database of Applications
  12. Closing Remarks and Open Problems

Key Result

Theorem 2.1

(Bézout cls97) The system $P({\bf x}) = {\bf 0}$ has no more than $D$ isolated solutions, counted with multiplicities.

Figures (11)

  • Figure 1: Newton polytopes $Q_1$, $Q_2$, a mixed subdivision of $Q_1 + Q_2$ with volumes.
  • Figure 3: The secant and tangent predictor with step length $h$.
  • Figure 4: Intersection of quadrics: a degenerate and a target configuration.
  • Figure 5: Triangulation of the Newton polytope of $P$ with polyhedral homotopy $\widehat{P}$.
  • Figure 6: In ${\Bbb P}^3$ two thick lines meet four given lines $L_1$, $L_2$, $L_3$, and $L_4$ in a point. At the left we see a special configuration and the general configuration is at the right.
  • ...and 6 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 5.4
  • Theorem 5.5
  • Theorem 5.6
  • Theorem 5.7