  We revisit the method of Carleman linearization for systems of ordinary
differential equations with polynomial right-hand sides. This transformation
provides an approximate linearization in a higher-dimensional space through the
exact embedding of polynomial nonlinearities into an infinite-dimensional
linear system, which is then truncated to obtain a finite-dimensional
representation with an additive error. To the best of our knowledge, no
explicit calculation of the error bound has been studied. In this paper, we
propose two strategies to obtain a time-dependent function that locally bounds
the truncation error. In the first approach, we proceed by iterative
backwards-integration of the truncated system. However, the resulting error
bound requires an a priori estimate of the norm of the exact solution for the
given time horizon. To overcome this difficulty, we construct a combinatorial
approach and solve it using generating functions, obtaining a local error bound
that can be computed effectively.
