Travaux récents de Ju. V. Nesterenko
Let K be an algebraically closed field, complete for an ultra- metric absolute value, let D be an infinite subset of K and let H(D) be the set of analytic elements on D. We denote by Mult(H(D), UD) the set of semi-norms Phi of the K-vector space H(D) which are continuous with respect to the topology of uniform convergence on D and which satisfy further Phi(f g)=Phi(f) Phi(g) whenever f,g elements of H(D) such that fg element of H(D). This set is provided with the topology of simple convergence....
In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V → k be the corresponding standard part map. Under a mild assumption on (R,V) we show that a definable set X ⊆ Vⁿ admits a triangulation that induces a triangulation of its standard part st X ⊆ kⁿ.
In this paper we investigate Hesse’s elliptic curves , and construct their twists, over quadratic fields, and over the Galois closures of cubic fields. We also show that is a twist of over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, , to parametrize all of quadratic fields and cubic ones. It should be noted that is a twist of as algebraic curves because it may not always have any rational points...
We present a simple proof of the theorem which says that for a series of extensions of differential fields K ⊂ L ⊂ M, where K ⊂ M is Picard-Vessiot, the extension K ⊂ L is Picard-Vessiot iff the differential Galois group is a normal subgroup of . We also present a proof that the probability function Erf(x) is not an elementary function.
A (monic) polynomial is called intersective if the congruence mod has a solution for all positive integers . Call nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group can be realized as the Galois group over of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable reduces to the ordinary...