Related papers: Hilbert's Tenth Problem for function fields of cha…

Given an elliptic curve ${\mathcal E}$ over a field $K$ it is a challenging problem to write down explicit elements of its endomorphism ring ${\rm End}({\mathcal E});$ the problem amounts to find all possible solutions to a functional…

We prove the geometric Bogomolov conjecture over a function field of characteristic zero.

We prove the undecidability of determining whether a Turing machine yields an eventually periodic trajectory. From this, we deduce the undecidability of orbit finiteness in the polynomial dynamical system on infinite tuples of integers.

We consider the entire characteristic functions of order 2 and we prove some decomposition theorems in a multidimensional case. We show that the lack of zeros of the density function is a necessary but not a sufficient (as in the…

We present an approach to defining Hilbert spaces of functions depending on infinitely many variables or parameters, with emphasis on a weighted tensor product construction based on stable space splittings, The construction has been used in…

We describe a method to perform functional operations on probability distributions of random variables. The method uses reproducing kernel Hilbert space representations of probability distributions, and it is applicable to all operations…

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.

We consider matrix problems in Hilbert spaces (orthoscalar representations of quivers and posets). A criterion of tameness of the problem of classification of indecomposable orthoscalar representations of a quiver is given.

A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if $\mbox{Cons}(\Sigma)$ is undecidable for every nonempty finite $\Sigma \subseteq \mbox{Th}(K; \mathcal{L})$. We extend a construction of Ziegler and (among other…

Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to…

We provide a framework for using elliptic curves with complex multiplication to determine the primality or compositeness of integers that lie in special sequences, in deterministic quasi-quadratic time. We use this to find large primes,…

We use Heegner points to prove the existence of nontorsion rational points on the elliptic curve $y^2 = x^3 + D$ for any rational number $D=a/b$ such that $a$ and $b$ are squarefree integers for which $6$, $a$, and $b$ are pairwise…

We prove that the jacobian of a hyperelliptic curve y^2=f(x) is absolutely simple if deg(f)=q+1 where q is a power prime congruent to 5 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois…

We show that elliptic curves whose Mordell-Weil groups are finitely generated over some infinite extensions of $\Q$, can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite…

We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the…

A great deal of recent activity has centered on the question of whether, for a given Hilbert function, there can fail to be a unique minimum set of graded Betti numbers, and this is closely related to the question of whether the associated…

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give…

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…

A `trinomial hyper surface' is defined in \S 1 below. In this article, I provide a supportive reasoning towards the fact that there can be examples of trinomial hyper surfaces (at least over fields of characteristic 2) for which the…

We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number…