In 1980, Christol, Kamae, Mendès France and Rauzy stated in [3] an important theorem in automata theory. This theorem links sequences recognized by automata and algebraic formal power series. In 1994, Bruyère, Hansel, Michaux and Villemaire extended this theorem with a logical link in [2]. With theses two articles, we can translate the property for a formal power series to be algebraic in combi...

the freiheitssatz of magnus for one-relator groups is one of the cornerstones of combinatorial group theory. in this short note which is mostly expository we discuss the relationship between the freiheitssatz and corre-sponding results in free power series rings over fields. these are related to results of schneerson not readily available in english. this relationship uses a faithful representa...

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...

The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q : formal, analytic, algebraic, of systems of equations of the same quality Q, assuming the existence of power series solutions of a weaker quality Q’ < Q : approximated, formal. The results are frequently used in commutative algebra and algebraic geometry. We present a systematic argum...

1: Properties of the shuffle product in positive characteristic suggest to consider a p−homogeneous form σ : Fp〈〈X1, . . . ,Xk〉〉 −→ Fp〈〈X1, . . . ,Xk〉〉 on the vector space Fp〈〈X1, . . . ,Xk〉〉 of formal power series in k free noncommuting variables. The form σ preserves rational elements in Fp〈〈X1, . . . ,Xk〉〉, algebraic series of Fp[[X]] and induces a bijection on the affine subspace 1+m of for...

Soittola’s theorem characterizes R+or N-rational formal power series in one variable among the rational formal power series with nonnegative coefficients. We present here a new proof of the theorem based on Soittola’s and Perrin’s proofs together with some new ideas that allows us to separate algebraic and analytic arguments. c © 2008 Elsevier B.V. All rights reserved.