Related papers: Diophantine sets of polynomials over number fields

A general construction yielding infinitely many families of $D(m^2)$-triples of triangular numbers is presented. Moreover, each triple obtained from this construction contains the same triangular number $T_n$.

In this paper we prove an existence theorem concerning linear forms of a given Diophantine type and apply it to study the structure of the spectrum of lattice exponents.

By following the same construction pattern which Martin Davis proposed in a 1968 paper of his, we have obtained six quaternary quartic Diophantine equations that candidate as `rule-them-all' equations: proving that one of them has only a…

For a nonzero integer n, a set of m distinct nonzero integers {a_1,a_2,...,a_m} such that a_i a_j + n is a perfect square for all 1 <= i < j <= m, is called a D(n)-m-tuple. In this paper, by using properties of so-called regular Diophantine…

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$…

Let $\S $ be an arbitrary subset of $R^n$ where $R$ is a domain with the field of fractions $\K$. Denote the ring of polynomials in $n$ variables over $\K$ by $\K[\x].$ The ring of integer-valued polynomials over $\S,$ denoted by…

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative…

The "Diophantine" property of the zeros of certain polynomials in the Askey scheme, recently discovered by Calogero and his collaborators, is explained, with suitably chosen parameter values, in terms of the summation theorem of…

We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition.

Given two seprable irreducible polynomials $P_1$ and $P_2$ over a filed $\mathbb{K}$. We show that the rings $\mathbb{K}[X]/(P_1^n)$ and $\mathbb{K}[X]/(P_2^n)$ are isomorphic if and only if their residue fields $\mathbb{K}[X]/(P_1)$ and…

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is…

We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one…

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…

We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.

We study some properties and perspectives of the Hurwitz series ring $H_R[[t]]$, for a commutative ring with identity $R$. Specifically, we provide a closed form for the invertible elements by means of the complete ordinary Bell…

In this paper, as an extension of the integer case, we define polynomial functions over the residue class rings of Dedekind domains, and then we give canonical representations and counting formulas for such polynomial functions. In…

We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can…

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

In this article it is determined which integral reflection representations of the symmetric groups and the primitive complex reflection groups of degree $2$ have rings of invariants which are isomorphic to polynomial rings.