The field of definition of function fields and a problem in the reducibility of polynomials in two variables
Introduction Let K be a number field, O the ring of integers of K. Suppose we are given a projective curve Z, and a morphism :Z pr(C) (where pr(C) is the projective line) such that and Z are both defined over K. We denote by K(Z) the field of functions of Z defined over K, and from this data we obtain a permutation representation T of the Galois group G(K(Z)^/K(Pr(C))) (where K(Z) is the normal closure of K(Z) over K(Pr(C) ). In Section 1 we investigate (for our needs) the combinatorial and group theoretical aspects of the situation where
Algebraic curves P(x) − Q(y) = 0 and functional equations
In this article, we give several conditions implying the irreducibility of the algebraic curve P(x) − Q(y) = 0, where P, Q are rational functions. We also apply the results obtained to the functional equations P(f) = Q(g) and P(f) = cP(g), where c ∈ ℂ. For example, we show that for a generic pair of rational functions P, Q the first equation has no non-constant solutions f, g meromorphic on ℂ whenever (deg P − 1)(deg Q − 1) ≥ 2.
ASYMPTOTIC PROPERTIES OF SOME CLASSES OF GENERALIZED FUNCTIONS
This paper studies the connection between the asymptotic and quasi-asymptotic properties at infinity of slowly increasing generalized functions with supports on the half-line and the asymptotic and quasi-asymptotic properties of the real parts of their Laplace and Fourier transforms in a neighborhood of the origin. The study is caried out in the scale of regularly varying self-similar functions. The results are applied to the study of the asymptotic properties of solutions of linear passive systems, and also to the study of the connection between Abel and Ces?ro convergence (with respect to a self-similar weight) of the Fourier-Stieltjes series of nonnegative measures. Bibliography: 13 titles.