Rational curves in the logarithmic multiplicative group

Integration of Rational Functions: Rational Computation of the Logarithmic Part

This formula is not satisfactory under a computing point of view because it introduces more algebraic quantities than necessary. The number P(a)/Q'(a) is called the residue of the root a of Q. We introduce the notion of multiplicity of a residue b that is the number of roots a having b as a residue. Trager (1976) introduced a new formula involving less algebraic numbers: let S(y) be the resulta...

Enumerating Rational Curves: the Rational Fibration Method

Curves with Innnite K-rational Geometric Fundamental Group

1 Rational Geometric Fundamental Groups Let K be a nitely generated eld with separable closure K s and absolute Galois group G K : By a curve C=K we always understand a smooth geometrically irreducible projective curve. Let F(C) be its function eld and let (C) be the Galois group of the maximal unramiied extension of F(C): We have the exact sequence 1 ! g (C) ! (C) ! G K ! 1 (?) where g (C) is ...

Rational Curves

Rational curves and splines are one of the building blocks of computer graphics and geometric modeling. Although a rational curve is more exible than its polynomial counterpart , many properties of polynomial curves are not applicable to it. For this reason it is very useful to know if a curve presented as a rational space curve has a polynomial parametrization. In this paper, we present an alg...

Curves with infinite K-rational geometric fundamental group

where Πg(C) is the geometric (profinite) fundamental group of C×Spec(Ks) (i.e. Πg(C) is equal to the Galois group of the maximal unramified extension of F (C)⊗Ks). This sequence induces a homomorphism ρC from GK to Out(Πg(C)) which is the group of automorphisms modulo inner automorphisms of Πg(C). It is well known that ρC is an important tool for studying C. For instance, it determines C up to ...