3,227 research outputs found
Hironaka's characteristic polygon and effective resolution of surfaces
Hironaka's concept of characteristic polyhedron of a singularity has been one
of the most powerful and fruitful ideas of the last decades in singularity
theory. In fact, since then combinatorics have become a major tool in many
important results. However, this seminal concept is still not enough to cope
with some effective problems: for instance, giving a bound on the maximum
number of blowing--ups to be performed on a surface before its multiplicity
decreases. This short note shows why such a bounding is not possible, at least
with the original definitions.Comment: 7 pages, 1 figur
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 simultaneous arithmetic progressions on elliptic curves
In this paper we study elliptic curves which have a number of points whose
coordinates are in arithmetic progression. We first motivate this diophantine
problem, prove some results, provide a number of interesting examples and,
finally point out open questions which focus on the most interesting aspects of
the problem for us.Comment: 22 page
- …