168 research outputs found
Computing Puiseux series : a fast divide and conquer algorithm
Let be a polynomial of total degree defined over
a perfect field of characteristic zero or greater than .
Assuming separable with respect to , we provide an algorithm that
computes the singular parts of all Puiseux series of above in less
than operations in , where
is the valuation of the resultant of and its partial derivative with
respect to . To this aim, we use a divide and conquer strategy and replace
univariate factorization by dynamic evaluation. As a first main corollary, we
compute the irreducible factors of in up to an
arbitrary precision with arithmetic
operations. As a second main corollary, we compute the genus of the plane curve
defined by with arithmetic operations and, if
, with bit operations
using a probabilistic algorithm, where is the logarithmic heigth of .Comment: 27 pages, 2 figure
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
Computing all integer solutions of a genus 1 equation
The Elliptic Logarithm Method has been applied with great successto the problem of computing all integer solutions of equations ofdegree 3 and 4 defining elliptic curves. We extend this methodto include any equation f(u,v)=0 that defines a curve of genus 1.Here f is a polynomial with integer coefficients and irreducible overthe algebraic closure of the rationals, but is otherwise of arbitrary shape and degree.We give a detailed description of the general features of our approach,and conclude with two rather unusual examples corresponding to equationsof degree 5 and degree 9.Elliptic curve;Elliptic logarithm;Dophantine equation
Computing low-degree factors of lacunary polynomials: a Newton-Puiseux approach
We present a new algorithm for the computation of the irreducible factors of
degree at most , with multiplicity, of multivariate lacunary polynomials
over fields of characteristic zero. The algorithm reduces this computation to
the computation of irreducible factors of degree at most of univariate
lacunary polynomials and to the factorization of low-degree multivariate
polynomials. The reduction runs in time polynomial in the size of the input
polynomial and in . As a result, we obtain a new polynomial-time algorithm
for the computation of low-degree factors, with multiplicity, of multivariate
lacunary polynomials over number fields, but our method also gives partial
results for other fields, such as the fields of -adic numbers or for
absolute or approximate factorization for instance.
The core of our reduction uses the Newton polygon of the input polynomial,
and its validity is based on the Newton-Puiseux expansion of roots of bivariate
polynomials. In particular, we bound the valuation of where is
a lacunary polynomial and a Puiseux series whose vanishing polynomial
has low degree.Comment: 22 page
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