47,416 research outputs found
Computational problems in the theory of Abelian groups
In this thesis, the worst-case time complexity bounds on the algorithms for the problems mentioned below have been improved.
A. Algorithms on abelian groups represented by a set of defining relations for computing:
(I) a canonical basis for finite abelian groups
(II) a canonical basis for Infinite abelian group
B. Algorithms for computing:
(I) Hermite normal form of an Integer matrix
(II) The Smith normal form of an Integer matrix
(III) The set of all solutions of a system of Diophantine Equations
C. Algorithms on abelian groups represented by an explicit set of generators for computing:
(I) the order of an element (space complexity 1s only improved)
(II) a complete basis for a finite abelian group
(III) membership-Inclusion testing
(IV) the union and Intersection of two finite abelian groups
D. A classification of the relative complexity of computational problems on abelian groups (as above) factorization and primility testing.
E. Algorithms on abelian subgroups of the symmetric group for computing:
(I) the complete structure of a group
(II) membership-Indus Ion testing
(III) the union of two abelian groups
(IV) the Intersection of two abelian groups
Fast Computation of Smith Forms of Sparse Matrices Over Local Rings
We present algorithms to compute the Smith Normal Form of matrices over two
families of local rings.
The algorithms use the \emph{black-box} model which is suitable for sparse
and structured matrices. The algorithms depend on a number of tools, such as
matrix rank computation over finite fields, for which the best-known time- and
memory-efficient algorithms are probabilistic.
For an \nxn matrix over the ring \Fzfe, where is a power of an
irreducible polynomial f \in \Fz of degree , our algorithm requires
\bigO(\eta de^2n) operations in \F, where our black-box is assumed to
require \bigO(\eta) operations in \F to compute a matrix-vector product by
a vector over \Fzfe (and is assumed greater than \Pden). The
algorithm only requires additional storage for \bigO(\Pden) elements of \F.
In particular, if \eta=\softO(\Pden), then our algorithm requires only
\softO(n^2d^2e^3) operations in \F, which is an improvement on known dense
methods for small and .
For the ring \ZZ/p^e\ZZ, where is a prime, we give an algorithm which
is time- and memory-efficient when the number of nontrivial invariant factors
is small. We describe a method for dimension reduction while preserving the
invariant factors. The time complexity is essentially linear in where is the number of operations in \ZZ/p\ZZ to evaluate the
black-box (assumed greater than ) and is the total number of non-zero
invariant factors.
To avoid the practical cost of conditioning, we give a Monte Carlo
certificate, which at low cost, provides either a high probability of success
or a proof of failure. The quest for a time- and memory-efficient solution
without restrictions on the number of nontrivial invariant factors remains
open. We offer a conjecture which may contribute toward that end.Comment: Preliminary version to appear at ISSAC 201
Computing Nilpotent Quotients in Finitely Presented Lie Rings
A nilpotent quotient algorithm for finitely presented Lie rings over Z
(LieNQ) is described. The paper studies graded and non-graded cases separately.
The algorithm computes the so-called nilpotent presentation for a finitely
presented, nilpotent Lie ring. The nilpotent presentation consists of
generators for the abelian group and the products---expressed as linear
combinations---for pairs formed by generators. Using that presentation the word
problem is decidable in . Provided that the Lie ring is graded, it is
possible to determine the canonical presentation for a lower central factor of
. LieNQ's complexity is studied and it is shown that optimizing the
presentation is NP-hard. Computational details are provided with examples,
timing and some structure theorems obtained from computations. Implementation
in C and GAP 3.5 interface is available.Comment: DVI and Post-Script files onl
- …