Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Kernel curves associated with walks in the quarter plane

Thomas Dreyfus, Charlotte Hardouin, Julien Roques, Michael F. Singer

Abstract

The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve, this defines. In the present paper, we investigate the basic properties of the kernel curve (irreducibility, singularities, genus, uniformization, etc).

On the Kernel curves associated with walks in the quarter plane

Abstract

The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve, this defines. In the present paper, we investigate the basic properties of the kernel curve (irreducibility, singularities, genus, uniformization, etc).

Paper Structure

This paper contains 6 sections, 20 theorems, 69 equations, 2 figures.

Table of Contents

  1. Nondegenerate walks
  2. Singularities and genus of the kernel curve
  3. Involutive automorphisms of the kernel curve
  4. Uniformization of the kernel curve
  5. Genus zero case
  6. Genus one case

Key Result

Proposition 1.2

A model of walk is degenerate if and only if at least one of the following holds:

Figures (2)

  • Figure 1: The maps $\iota_{1},\iota_{2}$ restricted to the kernel curve$\overline{E_t}$
  • Figure 2: An idealized representation of the uniformization map used in Proposition \ref{['prop:parameterizationcompauto']}. The left hand side represents the complex Riemann sphere and the right hand side the curve $\overline{E_t}$, seen as an abstract complex algebraic curve.

Theorems & Definitions (52)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Lemma 1.3
  • proof : Proof of Lemma \ref{['lem1']}
  • Lemma 1.4
  • proof : Proof of Lemma \ref{['lem2']}
  • Lemma 1.5
  • proof : Proof of Lemma \ref{['lem3']}
  • Remark 1.6
  • ...and 42 more