107,593 research outputs found
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
Formal Desingularization of Surfaces - The Jung Method Revisited -
In this paper we propose the concept of formal desingularizations as a
substitute for the resolution of algebraic varieties. Though a usual resolution
of algebraic varieties provides more information on the structure of
singularities there is evidence that the weaker concept is enough for many
computational purposes. We give a detailed study of the Jung method and show
how it facilitates an efficient computation of formal desingularizations for
projective surfaces over a field of characteristic zero, not necessarily
algebraically closed. The paper includes a generalization of Duval's Theorem on
rational Puiseux parametrizations to the multivariate case and a detailed
description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure
Finite automata and algebraic extensions of function fields
We give an automata-theoretic description of the algebraic closure of the
rational function field F_q(t) over a finite field, generalizing a result of
Christol. The description takes place within the Hahn-Mal'cev-Neumann field of
"generalized power series" over F_q. Our approach includes a characterization
of well-ordered sets of rational numbers whose base p expansions are generated
by a finite automaton, as well as some techniques for computing in the
algebraic closure; these include an adaptation to positive characteristic of
Newton's algorithm for finding local expansions of plane curves. We also
conjecture a generalization of our results to several variables.Comment: 40 pages; expanded version of math.AC/0110089; v2: refereed version,
includes minor edit
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
Integral D-Finite Functions
We propose a differential analog of the notion of integral closure of
algebraic function fields. We present an algorithm for computing the integral
closure of the algebra defined by a linear differential operator. Our algorithm
is a direct analog of van Hoeij's algorithm for computing integral bases of
algebraic function fields
Approximate computations with modular curves
This article gives an introduction for mathematicians interested in numerical
computations in algebraic geometry and number theory to some recent progress in
algorithmic number theory, emphasising the key role of approximate computations
with modular curves and their Jacobians. These approximations are done in
polynomial time in the dimension and the required number of significant digits.
We explain the main ideas of how the approximations are done, illustrating them
with examples, and we sketch some applications in number theory
- …