Related papers: The Diophantine problem for rings of exponential p…

We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$,…

Let R be a recursive subring of a number field. We show that recursively enumerable sets are diophantine for the polynomial ring R[Z].

Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and G\"odel's Second Incompleteness Theorem, it is…

The positive existential theories of the sets $M_n(\mathbb N)$ without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorithmically undecidable. In further sections some easier…

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 prove that there is no algorithm to decide whether a given integer vector is in the closure of the entropic cone $\overline{\Gamma_{n}^{*}}$. Equivalently, there is no decision procedure to determine whether a given integer-valued…

In this paper we will formulate $4\times4$ Riemann-Hilbert problems for Toeplitz+Hankel determinants and the associated system of orthogonal polynomials, when the Hankel symbol is supported on the unit circle and also when it is supported…

Let $K$ be a large field such that $K[\sqrt{-1}]$ is not algebraically closed and $F/K$ a function field in one variable. Extending techniques and results from earlier work with Becher and Dittmann, we show that every valuation ring on $F$…

We produce a fully faithful functor from finite type nilpotent spaces to cosimplicial binomial rings, thus giving an algebraic model of integral homotopy types. As an application, we construct an integral version of the…

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a…

We extend results of Denef, Zahidi, Demeyer and the second author to show the following. (1) Rational integers have a single-fold Diophantine definition over the ring of integral functions of any function field of characteristic 0. (2)…

We express the solutions to quadratic equations with two variables in the ring of integers using EDT0L languages. We use this to show that EDT0L languages can be used to describe the solutions to one-variable equations in the Heisenberg…

We investigate a Riemann-Hilbert problem (RHP), whose solution corresponds to a group of $q$-orthogonal polynomials studied earlier by Ismail et al. Using RHP theory we determine new asymptotic results in the limit as the degree of the…

To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth…

Let $A = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$. We will classify all the Gotzmann ideals of $A$ with at most $n$ generators. In addition, we will study Hilbert functions $H$ for which all homogeneous…

The DPRM (Davis-Putnam-Robinson-Matiyasevich) theorem is the main step in the negative resolution of Hilbert's 10th problem. Almost three decades of work on the problem have resulted in several equally surprising results. These include the…

Generalised Probabilistic Theories (GPTs) provide a unifying framework encompassing classical theories, quantum theories, as well as hypothetical alternatives. We investigate the problem of extending a system with a finite set of…

In this paper we study the Diophantine problem in Chevalley groups $G_\pi (\Phi,R)$, where $\Phi$ is an indecomposable root system of rank $> 1$, $R$ is an arbitrary commutative ring with $1$. We establish a variant of double centralizer…

Gorenstein rings are important to mathematical areas as diverse as algebraic geometry, where they encode information about singularities of spaces, and homotopy theory, through the concept of model categories. In consequence, the study of…

We show that variants of the classical reflection functors from quiver representation theory exist in any abstract stable homotopy theory, making them available for example over arbitrary ground rings, for quasi-coherent modules on schemes,…