First integrals and invariants of systems of ODEs
Mateja Grašič, Abdul Salam Jarrah, Valery G. Romanovski
TL;DR
This work develops algorithmic, symbolic methods to compute generators of the algebras of monomial and polynomial first integrals for $n$-dimensional autonomous systems with diagonal linear part, including when eigenvalues are algebraic over $\mathbb{Q}$. By linking monomial invariants to $Poincaré{-}Dulac$ normal forms through centralizers and invariant modules, the authors provide concrete procedures based on Hilbert bases, Smith normal form, and Gröbner bases to describe $I(\lambda)$ and the $C^{pol}(A)$-module structure. They introduce and implement algorithms (Algorithm 1–3) to compute Hilbert bases $H_\lambda$ and $H_L$, construct equivariants, and derive Stanley and Bruno decompositions of the normal-form module, with detailed examples illustrating the emergence of nontrivial invariants and higher-order equivariants. The methods generalize to algebraic eigenvalues and yield practical tools for classifying vector fields via invariants and normal forms, with potential impact on center–focus problems and symmetry-driven reductions.
Abstract
We investigate the interplay between monomial first integrals, polynomial invariants of certain group action, and the Poincaré-Dulac normal forms for autonomous systems of ODEs with diagonal matrix of the linear part. Using tools from computational algebra, we develop an algorithmic approach for identifying generators of the algebras of monomial and polynomial first integrals, which works in the general case where the matrix of the linear part includes algebraic complex eigenvalues. Our method also provides a practical tool for exploring the algebraic structure of polynomial invariants and their relation to the Poincaré-Dulac normal forms of the underlying vector fields.