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Certifying Galois/monodromy Actions via Homotopy Graphs

Timothy Duff, Kisun Lee

Abstract

We develop a certified numerical algorithm for computing Galois/monodromy groups of parametrized polynomial systems. Our approach employs certified homotopy path tracking to guarantee the correctness of the monodromy action produced by the algorithm, and builds on previous ``homotopy graph" frameworks. We conduct extensive experiments with an implementation of this algorithm, which we have used to certify properties of several notable Galois/monodromy groups which arise in several examples drawn from pure and applied mathematics.

Certifying Galois/monodromy Actions via Homotopy Graphs

Abstract

We develop a certified numerical algorithm for computing Galois/monodromy groups of parametrized polynomial systems. Our approach employs certified homotopy path tracking to guarantee the correctness of the monodromy action produced by the algorithm, and builds on previous ``homotopy graph" frameworks. We conduct extensive experiments with an implementation of this algorithm, which we have used to certify properties of several notable Galois/monodromy groups which arise in several examples drawn from pure and applied mathematics.
Paper Structure (16 sections, 4 theorems, 25 equations, 2 figures, 2 tables, 3 algorithms)

This paper contains 16 sections, 4 theorems, 25 equations, 2 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Interval arithmetic and the Krawczyk method
  4. Galois/monodromy groups of polynomial systems
  5. Certified path tracking
  6. Certified monodromy computation with homotopy graphs
  7. Parameter homotopies and edge correspondences
  8. Generating monodromy permutations from a homotopy graph
  9. Certified monodromy computation
  10. Experiments
  11. Simple univariate families
  12. A Belyi function for the Mathieu group
  13. 27 lines on a symmetric cubic surface
  14. Nearest point on an algebraic variety
  15. P3P: The Perspective-3-Point problem
  16. ...and 1 more sections

Key Result

Theorem 2.1

guillemot2024validated If $K(F,x,r,A)\subset r\rho B$ for some $\rho<1$, then the system $F(x)=0$ has a unique solution $x^*$ in the box $x+rB$, and $\|x-x^*\|\le r\rho$. Moreover, the quasi-Newton map $x\mapsto x-AF(x)$ is $\rho$-Lipschitz on $x+rB$.

Figures (2)

  • Figure 1: Illustration of certified monodromy computation for a $4$-to-$1$ projection $\pi:X\to\mathbb{C}^m$. The gray circle depicts the parameter space $\mathbb{C}^m$ with a homotopy graph (the base vertex $z_0$ is colored in blue). For each edge (a dashed line), certified path tracking lifts the parameter path to a curve on the incidence variety $X\subset \mathbb{C}^m\times\mathbb{C}^n$. This produces an edge correspondence, which is recorded to obtain monodromy permutations. Edges of the homotopy graph are colored once their correspondences have been computed; the colored curves indicate the corresponding certified lifts.
  • Figure 2: Homotopy graph $\mathcal{G}$ encircling branch points $z\in \{ 0,1\}$ for the Belyi family $f(x;g)=z$ defined by \ref{['eq:mathieu-polynomials']}, with $g$ satisfying \ref{['eq:g-poly']}.

Theorems & Definitions (10)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • Remark 3.4
  • Remark 3.5
  • Proposition 3.6
  • proof