On the structure of nilpotent normal form modules
The paper is concerned with the normal form theory of the differential equation \[ \dot{x}=Nx+a_2(x)+\cdots+a_j(x)+\cdots, \] where \(N\) is a nilpotent matrix, \(x\in\mathbb{R}^n\) and each \(a_j(x)\) is a homogeneous polynomial vector field of degree \(j\). An algorithm is presented that produces a Stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field given a Stanley decomposition for the ring of invariants. This reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm. Both the inner product normal form and the sl(2) normal form are covered within a unified theory, and simplified (nonequivariant) versions of each normal form are presented.
scientific article; zbMATH DE number 1797541