Computer Algebra Algorithms

@inproceedings{Kaltofen1987ComputerAA,
  title={Computer Algebra Algorithms},
  author={Erich L. Kaltofen},
  year={1987}
}
  • E. Kaltofen
  • Published 1 June 1987
  • Computer Science, Mathematics
INTRODUCTION ...................................................................................................................... 1 ARITHMETIC ........................................................................................................................... 2 Integer and Polynomial Addition, Multiplication, and Division with Remainder ............. 2 Integer and Polynomial Greatest Common Divisor ........................................................... 3 Floating Point Number s… 

Figures from this paper

Computing isomorphisms and embeddings of finite fields
TLDR
The goal is to describe algorithms to efficiently represent and evaluate one such embedding, unique up to F<sub><i>q</i></sub>-automorphisms of <i>k.</i.
Computer Algebra Software for Scientific Applications
TLDR
The astute user will realize that he has reached a threshold where qualitatively new tools for further proceeding are needed and to provide these tools is the realm of software engineering.
1 Major Research Results 1 . 1 Polynomial Factorization
In the following the EKbib and BASE URL is https:// users.cs.duke.edu/~elk27/bibliography/. The COURSEBASE URL is https://kaltofen.math.ncsu. edu/courses/. Many of my publications are accessible
Publications by Erich Kaltofen 1 Major Research Results 1.1 Polynomial Factorization 1.2 Linear Algebra
TLDR
In the following the BASE URL for the online document is http://www.math.ncsu.edu/~kaltofen/ bibliography, where y is the year of publication (last two digits) and l is the citation key in the BASE/kal tofen.bib file.
The “Seven Dwarfs” of Symbolic Computation
We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. We will
Parallelism in Hermite and Smith normal forms
The Smith and Hermite normal forms play an important role in various fields of investigations. In many applications it is crucial to compute the Smith or Hermite normal form of an integral matrix. A
A Scientific Basis for Computational Science
TLDR
This paper makes a case for an affirmative answer that relies on the concept of "generic scientific task" and argues that theoretical understanding is to be attained by identifying and automating such tasks.
Computational problems in the theory of finite fields
  • R. Lidl
  • Mathematics
    Applicable Algebra in Engineering, Communication and Computing
  • 2005
We give a survey of selected topics in the theory of finite fields with emphasis on computational aspects including recent advances and open problems.
Fast Algorithms for Towers of Finite Fields and Isogenies. (Algorithmes Rapides pour les Tours de Corps Finis et les Isogénies)
TLDR
Dans cette these nous appliquons des techniques provenant du calcul formel et de the theorie des langages afin d'ameliorer les operations elementaires dans certaines tours de corps finis d'accelerer l'algorithme de Couveignes.
Generic Tasks of Scientific Discovery
TLDR
This paper assemble a broad array of previous work as evidence to support the concept of generic scientific task as an organizing principle for research on scientific discovery, or more broadly, scientific inference.
...
...

References

SHOWING 1-10 OF 246 REFERENCES
The Calculation of Multivariate Polynomial Resultants
TLDR
An efficient algorithm is presented for the exact calculation of resultants of multivariate polynomials with integer coefficients over GF(p) using modular homomorphisms and the Chinese remainder theorem, and other algorithms are compared.
Computer Algebra
TLDR
This introduction gives a working definition of computer algebra, the organization of research activities in this field, and the overall structure and the intention of the present volume on computer algebra.
A new algorithm for factoring polynomials over finite fields
TLDR
A new probabilistic method is presented which, when combined with the above algorithms, avoids the need for both resultants and linear equations and leads to algorithms which are conceptually simpler than previous methods.
Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
TLDR
If Q is an inftmte field (e.g, the rauonal numbers or the complex numbers), then the equwalence problem for ~ is probabilistlcally decidable in polynomml time and the equivalence problem is NP-hard.
On the Application of Buchberger's Algorithm to Automated Geometry Theorem Proving
User-based integration software
TLDR
The software p a c k a g e t h a t i s d e s c r i b e d i s t h e Utah v e r s i o n (REDUCE INT) o f t h E o r i g i n a l Norman and Moore (1976) i m p l e m e n t a t I o n.
Factoring integers with elliptic curves
TLDR
This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2.
Analytic methods in the analysis and design of number-theoretic algorithms
  • E. Bach
  • Computer Science, Mathematics
  • 1985
This book makes a substantial contribution to the understanding of a murky area of number theory that is important to computer science, an area relevant to the design and analysis of number-theoretic
Polynomial-Time Reductions from Multivariate to Bi- and Univariate Integral Polynomial Factorization
TLDR
An algorithm is presented which reduces the problem of finding the irreducible factors of f in polynomial-time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomials, which implies the following theorem.
A Taxonomy of Problems with Fast Parallel Algorithms
  • S. Cook
  • Computer Science, Mathematics
    Inf. Control.
  • 1985
...
...