Related papers: Three counterexamples concerning the Northcott pro…
There exists an infinite family of examples of subsets of $\mathbb{F}_q^2$ with $q^{4/3}$ elements whose distance sets are not the whole of $\mathbb{F}_q$.
Let $E$ be an elliptic curve over a number field $K$ with at least one real embedding and $L$ be a finite extension of $K$. We generalize a result of Habegger to show that $L(E_{\text{tor}})$, the field generated by the torsion points of…
Motivated by Lang-Vojta's conjecture, we show that the set of dominant rational self-maps of an algebraic variety over a number field with only finitely many rational points in any given number field is finite by combining Amerik's theorem…
A field F is said to have the Bogomolov Property related to a height function h, if h(a) is either zero or bounded from below by a positive constant for all a in F. In this paper we prove that the maximal algebraic extension of a number…
Motivated by the Shilov boundaries of bounded symmetric domains we consider arbitrary CR-quadrics in a complex linear space (of finite dimension) that have a certain symmetry property. For these the non-affine local CR-automorphisms have a…
In 2013 P. Habegger proved the Bogomolov property for the field generated over Q by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions fixed by…
We give two examples of metric measure spaces satisfying the measure contraction property MCP(K,N) but having different topological dimensions at different regions of the space. The first one satisfies MCP(0,3) and contains a subset…
A class of rational functions characterized by some wonderful properties is studied. The properties that identify this class include simple algebra (their inverses can be expressed in radicals), simple topology (the total space of the…
We continue to study and present concrete examples in characteristic 2 of compound Du Val singularities defined over an algebraically closed field which have one dimensional singular loci but cannot be written as products (a rational double…
In this paper, we first present a classification theorem of infinite-dimensional simple Novikov algebras over an algebraically closed field with characteristic 0. Then we classify all the irreducible modules of a certain…
This paper studies the weak one-step closure and one-step closure properties concerning the structure of Scott closures. We deduce that every quasicontinuous domain has weak one-step closure and show that a quasicontinuous poset need not…
We prove the equivalence of the local property for an irreducible regular Dirichlet form and the Markov property for the Gaussian field associated with the Dirichlet form. Moreover we introduce a strong Markov property for Gaussian fields…
We discuss Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications to hyperelliptic jacobians are given. In particular, we improve some of our earlier results.
In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical…
We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this…
We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1-formality property of the space may be reinterpreted in terms…
Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(\alpha^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p :…
Let $R$ be a commutative ring, we say that $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property, if $I\subseteq \bigcup_{P\in\mathcal{A}}P$ for an ideal $I$ of $R$, then there exists $P\in\mathcal{A}$ such that $I\subseteq P$. We…
It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic…
Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in $\mathbb{P}_R^5$ and smooth cubic hypersurfaces in $\mathbb{P}_R^4$ which are not stably…