Related papers: Diophantine equations defined by binary quadratic …
In this article, we study the non-trivial primitive solutions of a certain type for the Diophantine equations $x^p+y^p=2^rz^p$ and $x^p+y^p=z^2$ of prime exponent $p$, $r \in \mathbb{N}$, over a totally real field $K$. Then for $r=2,3$, we…
We solve Diophantine equations of the type $ a \, (x^3 \!+ \! y^3 \!+ \! z^3 ) = (x \! + \! y \! + \! z)^3$, where $x,y,z$ are integer variables, and the coefficient $a\neq 0$ is rational. We show that there are infinite families of such…
An open problem of arithmetic Ramsey theory asks if given a finite $r$-colouring $c:\mathbb{N}\to\{1,...,r\}$ of the natural numbers, there exist $x,y\in \mathbb{N}$ such that $c(xy)=c(x+y)$ apart from the trivial solution $x=y=2$. More…
We introduce two Diophantine conditions on rotation numbers of interval exchange maps (i.e.m) and translation surfaces: the \emph{absolute Roth type condition} is a weakening of the notion of Roth type i.e.m., while the \emph{dual Roth…
For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…
We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…
We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$, $x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential functions are…
Rational double points are the simplest surface singularities. In this essay we will be mainly concerned with the geometry of the exceptional set corresponding to the resolution of a rational double point. We will derive the classification…
In this paper, we determine the isomorphic classes of Morita equivalent subalgebras of irrational rotation algebras. It is based on the solution of the quadratic Diophantine equations. We determine the irrational rotation algebras that have…
Fix a finite group $G$. We seek to classify varieties with $G$-action equivariantly birational to a representation of $G$ on affine or projective space. Our focus is odd-dimensional smooth complete intersections of two quadrics, relating…
Consider a form $g(x_1,...,x_s)$ of degree $d$, having coefficients in the completion $F_q((1/t))$ of the field of fractions $F_q(t)$ associated to the finite field $F_q$. We establish that whenever $s>d^2$, then the form $g$ takes…
We study some Diophantine problems related to triangles with two given integral sides. We solve two problems posed by Zolt\'an Bertalan and we also provide some generalization.
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)…
Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential…
Starting from Ritt's classical theorems, we give a survey of results in functional decomposition of polynomials and of applications in Diophantine equations. This includes sufficient conditions for the indecomposability of polynomials, the…
Using the modular method, we study solutions to the Diophantine equation $$Aa^p+Bb^p=Cc^2$$ over number fields. We first prove an asymptotic result for general number fields satisfying an appropriate $S$-unit condition by assuming some…
We study the solutions of a Diophantine equation of the form $a^x+b^y=c^z$, where $a\equiv 2 \pmod 4$, $b\equiv 3 \pmod 4$ and $\gcd (a,b,c)=1$. The main result is that if there exists a solution $(x,y,z)=(2,2,r)$ with $r>1$ odd then this…
Some new decidability results for multiplicative matrix equations over algebraic number fields are established. In particular, special instances of the so-called knapsack problem are considered. The proofs are based on effective methods for…
We consider the problem of Diophantine approximation on semisimple algebraic groups by rational points with restricted numerators and denominators and establish a quantitative approximation result for all real points in the group by…
We study systems of polynomial equations in several classes of finitely generated rings and algebras. For each ring $R$ (or algebra) in one of these classes we obtain an interpretation by systems of equations of a ring of integers $O$ of a…