Decomposition of integer-valued polynomial algebras

Let D be a commutative domain with field of fractions K, let A be a torsion-free D-algebra, and let B be the extension of A to a K-algebra. The set of integer-valued polynomials on A is $\mathrm{Int}(A)=\{f\in B[X]|f(A)?A\}$, and the intersection of $\mathrm{Int}(A)$ with $K[X]$ is ${\mathrm{Int}}_K(A)$, which is a commutative subring of $K[X]$. The set $\mathrm{Int}(A)$ may or may not be a ring, but it always has the structure of a left ${\mathrm{Int}}_K(A)$-module.A D-algebra A which is free as a D-module and of finite rank is called ${\mathrm{Int}}_K$-decomposable if a D-module basis for A is also an ${\mathrm{Int}}_K(A)$-module basis for $\mathrm{Int}(A)$; in other words, if $\mathrm{Int}(A)$ can be generated by ${\mathrm{Int}}_K(A)$ and A. A classification of such algebras has been given when D is a Dedekind domain with finite residue rings. In the present article, we modify the definition of ${\mathrm{Int}}_K$-decomposable so that it can be applied to D-algebras that are not necessarily free by defining A to be ${\mathrm{Int}}_K$-decomposable when $\mathrm{Int}(A)$ is isomorphic to ${\mathrm{Int}}_K(A){?}_{D}A$. We then provide multiple characterizations of such algebras in the case where D is a discrete valuation ring or a Dedekind domain with finite residue rings. In particular, if D is the ring of integers of a number field K, we show that an ${\mathrm{Int}}_K$-decomposable algebra A must be a maximal D-order in a separable K-algebra B, whose simple components have as center the same finite unramified Galois extension F of K and are unramified at each finite place of F. Finally, when both D and A are rings of integers in number fields, we prove that ${\mathrm{Int}}_K$-decomposable algebras correspond to unramified Galois extensions of K.     

On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.     

A counterexample to the local–global principle of linear dependence for Abelian varieties

Let A be an Abelian variety defined over a number field k. Let P be a point in $A(k)$ and let X be a subgroup of $A(k)$. Gajda and Kowalski asked in 2002 whether it is true that the point P belongs to X if and only if the point $(P\mathrm{mod}\mathfrak{p})$ belongs to $(X\mathrm{mod}\mathfrak{p})$ for all but finitely many primes $\mathfrak{p}$ of k. We provide a counterexample.     

A minimum degree condition forcing a digraph to be k-linked

Given a digraph D, let ${\delta }^0(D):=\min \{{\delta }^{+}(D),{\delta }^{-}(D)\}$ be the minimum semi-degree of D. In [D. Kühn and D. Osthus, Linkedness and ordered cycles in digraphs, submitted] we showed that every sufficiently large digraph D with ${\delta }^0(D)\ge n/2+\ell -1$ is ℓ-linked. The bound on the minimum semi-degree is best possible and confirms a conjecture of Manoussakis [Y. Manoussakis, k-linked and k-cyclic digraphs, J. Combinatorial Theory B 48 (1990) 216-226]. We [D. Kühn and D. Osthus, Linkedness and ordered cycles in digraphs, submitted] also determined the smallest minimum semi-degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every sequence $s_1,\dots ,s_k$ of distinct vertices of D there is a directed cycle which encounters $s_1,\dots ,s_k$ in this order.     

On the Gromov non-hyperbolicity of certain domains in Cn

In this paper, we prove that if Ω is a bounded convex domain in ${\mathbb{C}}^n$, $n\ge 2$, and S is an affine complex hyperplane such that $\mathrm{\Omega }\cap S$ is not empty, then $\mathrm{\Omega }?S$ is not Gromov hyperbolic with respect to the Kobayashi distance. Next, we show that if X is a bounded convex domain in ${\mathbb{C}}^n$, then $\mathrm{\Omega }=\{(z,w)\in X\times {\mathbb{C}}^{?},\left|w\right|<{\mathrm{e}}^{?\phi (z)}\}$ is not Gromov hyperbolic, where φ is a strictly plurisubaharmonic function on X continuous up to $\stackrel{￣}{X}$.