Related papers: Totally real Thue inequalities over imaginary quad…

We significantly improve our results of Glas. Mat., III. Ser. 53(2018), No. 2, 229-238, reducing relative Thue inequalities to absolute ones.

Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$ and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has…

We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $\alpha$ such that $K(\alpha)$ is a CM-field.…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ having at most one real conjugate and let $K$ be the algebraic number field ${\mathbf Q}(\alpha)$. For any unit $\epsilon$ of $K$ such that ${\mathbf Q}(\alpha\epsilon)=K$, we consider…

Let $F(x,y)$ be an irreducible form of degree $r\geq 3$ and having $s+1$ non-zero coefficients. Let $h\geq 1$ be an integer and consider the Thue inequality $$|F(x,y)|\leq h.$$ Following the seminal work of Thue in 1909, several papers were…

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary…

We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are…

In this paper, we study the number of integer pair solutions to the equation $|F(x,y)| = 1$ where $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible (over $\mathbb{Z}$) binary form with degree $n \geqslant 3$ and exactly three nonzero summands.…

We will use Thue-Siegel method, based on Pad\'e approximation via hypergeometric functions, to give upper bounds for the number of integral solutions to the equation $|F(x, y)| = 1$ as well as the inequalities $|F(x, y)| \leq h$, for a…

Let $\alpha$ be an algebraic number of degree $d\ge 3$ and let $K$ be the algebraic number field $\Q(\alpha)$. When $\varepsilon$ is a unit of $K$ such that $\Q(\alpha\varepsilon)=K$, we consider the irreducible polynomial $f_\varepsilon(X)…

Let $r,h\in\mathbb{N}$ with $r\geq 7$ and let $F(x,y)\in \mathbb{Z}[x ,y]$ be a binary form such that \[ F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r, \] where $\alpha$, $\beta$, $\gamma$ and $\delta$ are algebraic constants…

We revisit a work by R. Okazaki and prove that for every cubic binary form F(x, y) with large enough discriminant, the Thue equation |F(x, y)| = 1 has at most 7 solutions in integers x and y.

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb…

We will give an explicit upper bound for the number of solutions to cubic inequality |F(x, y)| \leq h, where F(x, y) is a cubic binary form with integer coefficients and positive discriminant D. Our upper bound is independent of h, provided…

The Thue-Siegel method is applied to derive an upper bound for the number of solutions to Thue's equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in this argument to handle…

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape $0<|F(x, y)| \leq h$, where $F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in…

We exactly determine the integral solutions to a previously untreated infinite family of cubic Thue equations of the form $F(x,y)=1$ with at least $5$ such solutions. Our approach combines elementary arguments, with lower bounds for linear…

Thue equations and their relative and inhomogeneous extensions are well known in the literature. There exist methods, usually tedious methods, for the complete resolution of these equations. On the other hand our experiences show that such…

Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a…

Let A be an arbitrary integral domain of characteristic 0 which is finitely generated over Z. We consider Thue equations $F(x,y)=b$ with unknowns x,y from A and hyper- and superelliptic equations $f(x)=by^m$ with unknowns from A, where the…