Related papers: Effective results for hyper- and superelliptic equ…

Let $f$ be a polynomial with coefficients in the ring $O_S$ of $S$-integers of a number field $K$, $b$ a non-zero $S$-integer, and $m$ an integer $\ge 2$. We consider the equation $( \star )$: $f(x) = b y^m$ in $x,y \in O_S$. Under the…

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…

Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer coefficients. We derive an upper bound for the number of integer points $(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box satisfying the equation…

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…

Consider the polynomial $f(x,y)=xy^k+C$ for $k\geq 2$ and any nonzero integer constant $C$. We derive an asymptotic formula for the $k$-free values of $f(x,y)$ when $x, y\leq H$. We also prove a similar result for the $k$-free values of…

We obtain effective bounds on the heights of algebraic integers whose orbits contain multiplicatively dependent values modulo S-integers. Our method is based on a new upper bound on the so-called S-height of polynomial values over the ring…

In this paper, it is shown that if F(x , y) is an irreducible binary form with integral coefficients and degree $n \geq 3$, then provided that the absolute value of the discriminant of F is large enough, the equation |F(x , y)| = 1 has at…

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 $F(x, y)$ 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…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

Given a polynomial $f$ defined over a number field $K$, we make effective certain special cases of a conjecture of S. Ih, on the finiteness of $f$-preperiodic points which are $S$-integral with respect to a fixed non-preperiodic point…

We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let $f = (f_1, \ldots, f_m)\subset \mathbb{Q}[y][x]$ with $y = (y_1, \ldots, y_t)$ and $x…

Let $\mathbb{K}$ be a field and let $f,g \in \mathbb{K}[x,y]$ be such that the ideal $\langle f,g \rangle$ is zero-dimensional. We study the Sylvester and B\'{e}zout resultant polynomial matrices, built by interpreting $f$ and $g$ as…

When $A=3$, the positive integral solutions of the so-called Markoff equation $$M_A:x^2 + y^2 + z^2 = Axyz$$ can be generated from the single solution $(1,1,1)$ by the action of certain automorphisms of the hypersurface. Since Markoff's…

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…

Let $k$ be a number field and $S$ a finite set of places of $k$ containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of $S$-integers of $k$. Moreover, we give an…

We give upper bounds for the number of integral solutions of bounded height to a system of equations $f_i(x_1,\ldots,x_n) = 0$, $1 \leq i \leq r$, where the $f_i$ are polynomials with integer coefficients. The estimates are obtained by…

We extend the notion of polynomial integration over an arbitrary circle $C$ in the Euclidean geometry over general fields $\mathbb F$ of characteristic zero as a normalized $\mathbb F$-linear functional on $\mathbb{F}\left[\alpha_1,…

Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime.…

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…