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On the degeneration of Kovalevskaya exponents of Laurent series solutions of quasi-homogeneous vector fields

Hayato Chiba

TL;DR

This work develops a systematic method to derive non-principle Laurent series solutions from principle ones for a quasi-homogeneous vector field $F$ that commutes with a vector field $G$. By analyzing Laurent expansions, indicial loci, and Kovalevskaya exponents via the K-matrix, the authors show how the flow of the $G$-driven parameters deforms the indicial data, yielding lower-dimensional solution families. A key contribution is establishing how the Kovalevskaya exponents of non-principle solutions are related to those of principle solutions, including a scaling by the degree $oldsymbol{ abla}$ and the explicit construction of lower indicial loci (Theorem 4.12) under broad conditions. The framework integrates quasi-homogeneity, commutation, and flows of free parameters to connect Painlevé-type integrable structures with the geometry of Laurent-series solution spaces, offering a systematic path to degenerations and a potential tool for singularity analysis in differential systems.

Abstract

A structure of families of Laurent series solutions of a quasi-homogeneous vector field is studied, where a given vector field is assumed to have a commutable vector field. For an $m$ dimensional vector field, a family of Laurent series solutions is called principle if it includes $m$ arbitrary parameters, and called non-principle if the number is smaller than $m$. Starting from a principle Laurent series solutions, a systematic method to obtain a non-principle Laurent series solutions is given. In particular, from the Kovalevskaya exponents of the principle Laurent series solutions, which is one of the invariants of quasi-homogeneous vector fields, the Kovalevskaya exponents of the non-principle Laurent series solutions are obtained by using the commutable vector field.

On the degeneration of Kovalevskaya exponents of Laurent series solutions of quasi-homogeneous vector fields

TL;DR

This work develops a systematic method to derive non-principle Laurent series solutions from principle ones for a quasi-homogeneous vector field that commutes with a vector field . By analyzing Laurent expansions, indicial loci, and Kovalevskaya exponents via the K-matrix, the authors show how the flow of the -driven parameters deforms the indicial data, yielding lower-dimensional solution families. A key contribution is establishing how the Kovalevskaya exponents of non-principle solutions are related to those of principle solutions, including a scaling by the degree and the explicit construction of lower indicial loci (Theorem 4.12) under broad conditions. The framework integrates quasi-homogeneity, commutation, and flows of free parameters to connect Painlevé-type integrable structures with the geometry of Laurent-series solution spaces, offering a systematic path to degenerations and a potential tool for singularity analysis in differential systems.

Abstract

A structure of families of Laurent series solutions of a quasi-homogeneous vector field is studied, where a given vector field is assumed to have a commutable vector field. For an dimensional vector field, a family of Laurent series solutions is called principle if it includes arbitrary parameters, and called non-principle if the number is smaller than . Starting from a principle Laurent series solutions, a systematic method to obtain a non-principle Laurent series solutions is given. In particular, from the Kovalevskaya exponents of the principle Laurent series solutions, which is one of the invariants of quasi-homogeneous vector fields, the Kovalevskaya exponents of the non-principle Laurent series solutions are obtained by using the commutable vector field.
Paper Structure (7 sections, 102 equations, 1 figure, 1 table)

This paper contains 7 sections, 102 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Settings and the Kovalevskaya exponents
  3. Properties of the vector field $G$
  4. Flow of the free parameters
  5. $\gamma =1$
  6. $\gamma \geq 2$
  7. The proof of Proposition 4.8.

Figures (1)

  • Figure 1: A schematic view of the flow of $F$ on the compactified space. The original phase space ${\mathbb C}^4$ is compactified by attaching $D$ at "infinity". There are three fixed points on $D$ that correspond to three indicial loci. The red dotted orbit indicates an orbit of the vector field $F+G/(\varepsilon +k_1)$ given in Proposition 4.5.