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

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.

💡 Research Summary

The paper investigates families of Laurent series solutions of quasi‑homogeneous polynomial vector fields, focusing on the relationship between “principal” families (containing the full set of m arbitrary parameters for an m‑dimensional system) and “lower” families (containing fewer parameters). The central idea is that, when a quasi‑homogeneous vector field F admits another quasi‑homogeneous vector field G that commutes with F, the flow generated by G can be used to move a principal Laurent series solution onto the boundary of its parameter manifold, thereby producing a lower‑dimensional family of solutions.

The authors begin by formalising the setting. Both F = (f₁,…,f_m) and G = (g₁,…,g_m) are assumed to satisfy three conditions: (A1) they share a weight vector (a₁,…,a_m) with degree 1 for F and degree γ ∈ ℕ for G; (A2) the Lie bracket