On the Galois spectra of polynomials with integral parameters.
On the Galois theory of difference fields.
On the Galois Theory of function fields of one variable over number fields.
On the Gauss-Lucas'lemma in positive characteristic
If is a polynomial with coefficients in the field of complex numbers, of positive degree , then has at least one root a with the following property: if , where is the multiplicity of , then (such a root is said to be a "free" root of ). This is a consequence of the so-called Gauss-Lucas'lemma. One could conjecture that this property remains true for polynomials (of degree ) with coefficients in a field of positive characteristic (Sudbery's Conjecture). In this paper it is shown that,...
On the geometrization of a lemma of Singer and van der Put
In this paper, we give a geometrization and a generalization of a lemma of differential Galois theory, used by Singer and van der Put in their reference book. This geometrization, in addition of giving a nice insight on this result, offers us the opportunity to investigate several points of differential algebra and differential algebraic geometry. We study the class of simple Δ-schemes and prove that they all have a coarse space of leaves. Furthermore, instead of considering schemes endowed with...
On the greatest common divisor of two univariate polynomials, II
On the group .
On the guaranteed convergence of the Japanese zero-finding method.
On the Győry-Sárközy-Stewart conjecture in function fields
We consider function field analogues of the conjecture of Győry, Sárközy and Stewart (1996) on the greatest prime divisor of the product for distinct positive integers , and . In particular, we show that, under some natural conditions on rational functions , the number of distinct zeros and poles of the shifted products and grows linearly with if . We also obtain a version of this result for rational functions over a finite field.
On the Hasse norm principle.
On the height of the Sylvester resultant.
On the homology of algebraic varieties over real closed fields.
On the index theorem for -adic differential equations. III. (Sur le théorème de l'indice des équations différentielles -adiques. III.)
On the infinite fern of Galois representations of unitary type
Let be a CM number field, an odd prime totally split in , and let be the -adic analytic space parameterizing the isomorphism classes of -dimensional semisimple -adic representations of satisfying a selfduality condition “of type ”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in has dimension at least . As important steps, and in any rank, we prove that any first order...
On the ...-invariant for quadratics forms and the linkage of cyclic algebras.
On the inverse problem of Galois theory.
The problem of the construction of number fields with Galois group over Q a given finite groups has made considerable progress in the recent years. The aim of this paper is to survey the current state of this problem, giving the most significant methods developed in connection with it.
On the irreducibility of 0,1-polynomials of the form f(x)xⁿ + g(x)
If f(x) and g(x) are relatively prime polynomials in ℤ[x] satisfying certain conditions arising from a theorem of Capelli and if n is an integer > N for some sufficiently large N, then the non-reciprocal part of f(x)xⁿ + g(x) is either identically ±1 or is irreducible over the rationals. This result follows from work of Schinzel in 1965. We show here that under the conditions that f(x) and g(x) are relatively prime 0,1-polynomials (so each coefficient is either 0 or 1) and f(0) = g(0) = 1, one...
On the irreducibility of certain polynomials with coefficients as products of terms in an arithmetic progression
On the irreducibility of the generalized Laguerre polynomials
On the Jacobian conjecture and the configurations of roots.