6 research outputs found
Integral points on generic fibers
Let P(x,y) be a rational polynomial and k in Q be a generic value. If the
curve (P(x,y)=k) is irreducible and admits an infinite number of points whose
coordinates are integers then there exist algebraic automorphisms that send
P(x,y) to the polynomial x or to x^2-dy^2. Moreover for such curves (and
others) we give a sharp bound for the number of integral points (x,y) with x
and y bounded.Comment: 12 page
Markoff-Rosenberger triples in geometric progression
Solutions of the Markoff-Rosenberger equation ax^2+by^2+cz^2 = dxyz such that
their coordinates belong to the ring of integers of a number field and form a
geometric progression are studied.Comment: To appear in Acta Mathematica Hungaric
Markoff-Rosenberger triples in arithmetic progression
We study the solutions of the Rosenberg--Markoff equation ax^2+by^2+cz^2 =
dxyz (a generalization of the well--known Markoff equation). We specifically
focus on looking for solutions in arithmetic progression that lie in the ring
of integers of a number field. With the help of previous work by Alvanos and
Poulakis, we give a complete decision algorithm, which allows us to prove
finiteness results concerning these particular solutions. Finally, some
extensive computations are presented regarding two particular cases: the
generalized Markoff equation x^2+y^2+z^2 = dxyz over quadratic fields and the
classic Markoff equation x^2+y^2+z^2 = 3xyz over an arbitrary number field.Comment: To appear in Journal of Symbolic Computatio
On the simplest sextic fields and related Thue equations
We consider the parametric family of sextic Thue equations where
is an integer and is a divisor of . We
show that the only solutions to the equations are the trivial ones with
.Comment: 12 pages, 2 table
Rational Bézier curves with infinitely many integral points
summary:In this paper we consider rational Bézier curves with control points having rational coordinates and rational weights, and we give necessary and sufficient conditions for such a curve to have infinitely many points with integer coefficients. Furthermore, we give algorithms for the construction of these curves and the computation of theirs points with integer coefficients
Some New Symbolic Algorithms for the Computation of Generalized Asymptotes
We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.Agencia Estatal de Investigació