2000 Mathematics Subject Classification: 12D10.We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its s-th derivative are realized by real polynomials.

Explicit monoid structure is provided for the class of canonical subfield preserving polynomials over finite fields. Some classical results and asymptotic estimates will follow as corollaries.

We study geodesic completeness for left-invariant Lorentz metrics on solvable Lie groups.

We define a notion of volume for sets definable in an o-minimal structure on an archimedean real closed field. We show that given a parametric family of continuous functions on the positive cone of an archimedean real closed field definable in an o-minimal structure, the set of parameters where the integral of the function converges is definable in the same structure.

Let α, β and γ be algebraic numbers of respective degrees a, b and c over ℚ such that α + β + γ = 0. We prove that there exist algebraic numbers α₁, β₁ and γ₁ of the same respective degrees a, b and c over ℚ such that α₁ β₁ γ₁ = 1. This proves a previously formulated conjecture. We also investigate the problem of describing the set of triplets (a,b,c) ∈ ℕ³ for which there exist finite field extensions K/k and L/k (of a fixed field k) of degrees a and b, respectively, such that the degree of the...

Let ${\mathbb{Z}}_d\wr$m$bethegeneralizedalternatinggroup.Weprovethatalldoublecoversof$ℤd ≀ m$occurasGaloisgroupsoveranyalgebraicnumberfield.WefurtherrealizesomeofthesedoublecoversastheGaloisgroupsofregularextensionsof\mathbb{Q}\left(T\right).Ifdisoddandm>7,theneverycentralextensionof$ℤd ≀ m$occursastheGaloisgroupofaregularextensionof\mathbb{Q}\left(T\right).Wefurtherimprovesomeofourearlierresultsconcerningdoublecoversofthegeneralizedsymmetricgroup$ℤd ≀ m$.$