2,348 research outputs found
Single-factor lifting and factorization of polynomials over local fields
Let f (x) be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of f (x), with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this ‘‘single-factor lifting’’ routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is requiredPostprint (published version
Genetics of polynomials over local fields
Let be a discrete valued field with valuation ring \oo, and let
\oo_v be the completion of \oo with respect to the -adic topology. In
this paper we discuss the advantages of manipulating polynomials in \oo_v[x]
in a computer by means of OM representations of prime (monic and irreducible)
polynomials. An OM representation supports discrete data characterizing the
Okutsu equivalence class of the prime polynomial. These discrete parameters are
a kind of DNA sequence common to all individuals in the same Okutsu class, and
they contain relevant arithmetic information about the polynomial and the
extension of that it determines.Comment: revised according to suggestions by a refere
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
FORM version 4.0
We present version 4.0 of the symbolic manipulation system FORM. The most
important new features are manipulation of rational polynomials and the
factorization of expressions. Many other new functions and commands are also
added; some of them are very general, while others are designed for building
specific high level packages, such as one for Groebner bases. New is also the
checkpoint facility, that allows for periodic backups during long calculations.
Lastly, FORM 4.0 has become available as open source under the GNU General
Public License version 3.Comment: 26 pages. Uses axodra
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
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