Principally tame regular sequences associated with the fourth Painlevé hierarchy with a large parameter (Q930130)

The aim of this paper is to establish some criteria for systems of algebraic equations admitting finite number of solutions. In the previous paper (see [\textit{T. Aoki} and \textit{N. Honda}, Regular sequence associated with the Noumi-Yamada equations with a large parameter, Algebraic Analysis of Differential Equations, Springer, 45--53 (2008)] the authors introduce the notion of tame regularity for systems of algebraic equations (or polynomials). It guarantees finiteness of the number of solutions if the number of equations coincides with the dimension of the base space. It turns out that a sequence of polynomials is tame regular if the sequence of principal homogeneous parts of these polynomials is tame regular. In this paper, the authors prove that the same statement is true for any standard ranking of \(C[x_1,\dots,x_n]\). This generalization permits to prove the existence and the finiteness of the leading terms of formal solutions to a general member of the fourth Painlevé hierarchy with a large parameter. Once the leading terms are determined, the higher order terms of the formal solutions can be obtained successively under the condition that the Jacobi matrix of the system never vanishes.