New convergence results for Carleman linearization

We prove new error bounds for finite Carleman truncations of polynomial ordinary differential equations. The analysis works directly in the original monomial basis and for selected observables, such as state coordinates. Using a Dyson–Duhamel expansion, we separate the degree-preserving linear part from the degree-raising nonlinear part and track how truncation errors can propagate back to the observable. The resulting bounds are degree-aware and retain logarithmic-norm information from the original linear dynamics. We obtain explicit finite-degree estimates and geometric convergence over certified time horizons. Comparisons with existing bounds, in particular those of Forets–Pouly, are given on the Stuart–Landau and Van der Pol systems.