1,072 research outputs found
Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]
We provide an irreducibility test in the ring K[[x]][y] whose complexity is
quasi-linear with respect to the valuation of the discriminant, assuming the
input polynomial F square-free and K a perfect field of characteristic zero or
greater than deg(F). The algorithm uses the theory of approximate roots and may
be seen as a generalization of Abhyankhar's irreducibility criterion to the
case of non algebraically closed residue fields. More generally, we show that
we can test within the same complexity if a polynomial is pseudo-irreducible, a
larger class of polynomials containing irreducible ones. If is
pseudo-irreducible, the algorithm computes also the valuation of the
discriminant and the equisingularity types of the germs of plane curve defined
by F along the fiber x=0.Comment: 51 pages. Title modified. Slight modifications in Definition 5 and
Proposition 1
Panorama of p-adic model theory
ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud
Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud
des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud
par une revue de la bibliographie
On Termination of Integer Linear Loops
A fundamental problem in program verification concerns the termination of
simple linear loops of the form x := u ; while Bx >= b do {x := Ax + a} where x
is a vector of variables, u, a, and c are integer vectors, and A and B are
integer matrices. Assuming the matrix A is diagonalisable, we give a decision
procedure for the problem of whether, for all initial integer vectors u, such a
loop terminates. The correctness of our algorithm relies on sophisticated tools
from algebraic and analytic number theory, Diophantine geometry, and real
algebraic geometry. To the best of our knowledge, this is the first substantial
advance on a 10-year-old open problem of Tiwari (2004) and Braverman (2006).Comment: Accepted to SODA1
Computation of Integral Bases
Let be a Dedekind domain, the fraction field of , and
a monic irreducible separable polynomial. For a given non-zero prime ideal
of we present in this paper a new method to compute a
-integral basis of the extension of determined by . Our
method is based on the use of simple multipliers that can be constructed with
the data that occurs along the flow of the Montes Algorithm. Our construction
of a -integral basis is significantly faster than the similar
approach from and provides in many cases a priori a triangular basis.Comment: 22 pages, 4 figure
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