It is known that Siegel's theorem on integral points is effective for Galois coverings of the projective line. In this paper we obtain a quantitative version of this result, giving an explicit upper bound for the heights of S-integral K-rational points in terms of the number field K, the set of places S and the defining equation of the curve. Our main tools are Baker's theory of linear forms in logarithms and the quantitative Eisenstein theorem due to Schmidt, Dwork and van der Poorten.

Let $f(x) = \Sigma a_ix^i$ be a monic polynomial of degree $n$ whose coefficients are algebraically independent variables over a base field $k$ of characteristic 0. We say that a polynomial $g(x)$ is generating (for the symmetric group) if it can be obtained from $f(x)$ by a nondegenerate Tschirnhaus transformation. We show that the minimal number ${\rm d}_k(n)$ of algebraically independent coefficients of such a polynomial is at least $[n/2]$. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert's 13th problem.

Our approach to this question (and generalizations) is based on the idea of the ‘essential dimension’ of a finite group $G$: the smallest possible dimension of an algebraic $G$-variety over $k$ to which one can ‘compress’ a faithful linear representation of $G$. We show that ${\rm d}_k(n)$ is just the essential dimension of the symmetric group ${\rm S}_n$. We give results on the essential dimension of other groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

The discriminant function $\Delta$ is a certain rigid analytic modular form defined on Drinfeld's upper half-plane $\Omega$. Its absolute value $\vert \Delta\vert$ may be considered as a function on the associated Bruhat–Tits tree ${\cal T}$. We compare $\log \vert \Delta\vert$ with the conditionally convergent complex-valued Eisenstein series $E$ defined on ${\cal T}$ and thereby obtain results about the growth of $\vert \Delta$ and of some related modular forms. We further determine to what extent roots may be extracted of $\Delta(z)/\Delta(nz)$, regarded as a holomorphic function on $\Omega$. In some cases, this enables us to calculate cuspidal divisor class groups of modular curves.

In this paper we construct smooth irreducible space curves $C$ which link geometrically by surfaces of minimal degree containing $C$ to curves $\Gamma$ of generic embedding dimension three. This produces interesting behavior with respect to both $C$ and $\Gamma$. The curves $\Gamma$ link to smooth connected curves by surfaces of low degree but cannot link to smooth connected curves by surfaces of high degree. The curves $C$ give counterexamples to a conjecture of Martin-Deschamps and Perrin.

A relation between the Euler characteristics of the Milnor fibres of a real analytic function is derived from a simple identity involving complex monodromy and complex conjugation. A corollary is the result of Coste and Kurdyka that the Euler characteristic of the local link of an irreducible algebraic subset of a real algebraic set is generically constant modulo 4. A similar relation for iterated Milnor fibres of ordered sets of functions is used to define topological invariants of ordered collections of algebraic subsets.

In this paper we solve the problem of desingularization of an absolutely isolated singularity of a differential equation, including the dicritical case. As an application, we prove the finiteness of the number of dicritical points in the blowing up tree of an absolutely isolated singularity.