Our aim is to prove that two formal power series of importance to quantum topology are Gevrey. These series are the Kashaev invariant of a knot (reformulated by Huynh and the second author) and the Gromov norm of the LMO of an integral homology 3-sphere. It follows that the power series associated to a simple Lie algebra and a homology sphere is Gevrey. Contrary to the case of analysis, our for...

Given a first order autonomous algebraic ordinary differential equation, we present a method to compute all formal series solutions. Furthermore, when the ground field is the field of the complex numbers, the computed formal power series solutions are indeed convergent in suitable neighborhoods. keywordAlgebraic autonomous differential equation, algebraic curve, local parametrization, formal po...

It is not possible to determine from the Hilbert series whether a graded Noetherian algebra is a complete intersection. Nevertheless, such Hilbert series satisfy very stringent conditions and we define a concept CI-type for formal power series, that embodies some of these necessary properties. This definition works for algebras that are not standard i. e. not generated in degree 1. For the clas...

We consider the question of whether the famous Parikh's theorem holds with multiplicities i.e., for formal power series instead of languages. We introduce two families of semilinear formal power series over a commutative product monoid. The strict hierarchy of algebraic, rational, recognizable and semi-linear formal power series is proved and in this way it is established that the Parikh's theo...