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On Hilbert's 16th Problem

Lars Andersen

Abstract

We prove that to each real singularity $f: (\mathbb{R}^{n}, 0) \to (\mathbb{R}^k, 0)$ with $k\geq 2$ one can associate systems of differential equations $\mathfrak{g}^{k}_f$ which are pushforwards in the category of $\mathcal{D}$-modules over $\mathbb{R}^{k}$ of the sheaf of real analytic functions on the total space of the Milnor fibration. We then use this to study Hilbert's 16th problem on polynomial dynamical systems in the plane.

On Hilbert's 16th Problem

Abstract

We prove that to each real singularity with one can associate systems of differential equations which are pushforwards in the category of -modules over of the sheaf of real analytic functions on the total space of the Milnor fibration. We then use this to study Hilbert's 16th problem on polynomial dynamical systems in the plane.
Paper Structure (11 sections, 3 theorems, 23 equations)

This paper contains 11 sections, 3 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. A Real Milnor Fibration
  3. Real Vanishing Cycles
  4. Notation
  5. $\mathcal{D}$-modules over the real numbers
  6. The Gauss-Manin Connection
  7. Real Analytic Solutions
  8. Polynomial Dynamical Systems in the Plane
  9. Hilbert's 16th Problem
  10. Localising a dynamical system
  11. The Solution

Key Result

Theorem 1

There exists a ring isomorphism

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1
  • proof
  • Definition 2
  • Definition 3
  • Corollary 1.1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • ...and 4 more