| Abstract: |
| In this talk, we study the relationship between monomial first integrals, polynomial invariants arising from certain group actions, and the Poincar\`e--Dulac normal forms of autonomous systems of ordinary differential equations with a diagonal linear part. Using tools from computational algebra, we develop algorithmic methods for identifying generators of the algebras of monomial and polynomial first integrals, including the general case where the eigenvalues of the linear part are algebraic complex numbers. The approach also provides a practical way to explore the structure of polynomial invariants and their connection to the Poincar\`e--Dulac normal forms of the underlying vector fields. |
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