A constructive version of Newton-Puiseux theorem for computing the Puiseux expansion of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regular...

Abstract. We give a polynomial-time algorithm of computing the classical Hurwitz numbers Hg,d, which were defined by Hurwitz 125 years ago. We show that the generating series of Hg,d for any fixed g > 2 lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the ...

where x is a complex variable and A(x) a square matrix of dimension n the entries of which are formal meromorphic power series. Write A = x(A0 +A1x+ · · ·) (A0 6= 0) for the series expansion of A, where the coefficients are matrices over a subfield K of the field of complex numbers. There exists a basis of n formal solutions of the form (see, e.g. Turritin, 1955; Wasow, 1967) yi(t) = etzi(t) (i...

Formal power series are an extension of formal languages. Recognizable formal power series can be captured by the so-called weighted finite automata, generalizing finite state machines. In this paper, motivated by codings of formal languages, we introduce and investigate two types of transformations for formal power series. We characterize when these transformations preserve recognizability, ge...

Formal power series are an extension of formal languages. Recognizable formal power series can be captured by the so-called weighted finite automata, generalizing finite state machines. In this paper, motivated by codings of formal languages, we introduce and investigate two types of transformations for formal power series. We characterize when these transformations preserve rationality, genera...

One of the main virtues of trees is the representation of formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in power series rings. When analyzed in terms of combinatorial Hopf algebras, the simplest examples yield interesting algebraic identities or enumerative...

The class of 2-automatic paperfolding sequences corresponds to the class of ultimately periodic sequences of unfolding instructions. We rst show that a paper-folding sequence is automatic ii it is 2-automatic. Then we provide families of minimal nite-state automata, minimal uniform tag sequences and minimal substitutions describing automatic paperfolding sequences, as well as a family of algebr...

The problem of algebraic dependence solutions to (non-linear) first order autonomous equations over an algebraically closed field characteristic zero is given a ‘complete’ answer, obtained independently model theoretic results on differentially fields. Instead, the geometry curves and generalized Jacobians provides key ingredient. Classification formal are treated. applied answer question $...

Abstract Let K be a finite field, ( x ) the field of rational functions in over and K formal power series . We show that under certain conditions integral combinations with algebraic coefficients U 1 -number are m -numbers , where is degree extension ), determined by these c...