A computer algebra user interface manifesto

Many computer algebra systems have more than 1000 built-in functions, making expertise difficult. Using mock dialog boxes, this article describes a proposed interactive general-purpose wizard for organizing optional transformations and allowing easy fine grain control over the form of the result even by amateurs. This wizard integrates ideas including: * flexible subexpression selection; * complete control over the ordering of variables and commutative operands, with well-chosen defaults; * interleaving the choice of successively less main variables with applicable function choices to provide detailed control without incurring a combinatorial number of applicable alternatives at any one level; * quick applicability tests to reduce the listing of inapplicable transformations; * using an organizing principle to order the alternatives in a helpful manner; * labeling quickly-computed alternatives in dialog boxes with a preview of their results, * using ellipsis elisions if necessary or helpful; * allowing the user to retreat from a sequence of choices to explore other branches of the tree of alternatives or to return quickly to branches already visited; * allowing the user to accumulate more than one of the alternative forms; * integrating direct manipulation into the wizard; and * supporting not only the usual input-result pair mode, but also the useful alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer Algebr

Symbolic computation: systems and applications

The article presents an overview of symbolic computation systems, their classification-in-history, the most popular CAS, examples of systems and some of their applications. Symbolics versus numeric, enhancement in mathematics, computing nature of CAS, related projects, networks, references are discussed

Macsyma: A personal history

AbstractThe Macsyma system arose out of research on mathematical software in the AI group at MIT in the 1960s. Algorithm development in symbolic integration and simplification arose out of the interest of people, such as the author, who were also mathematics students. The later development of algorithms for the GCD of sparse polynomials, for example, arose out of the needs of our user community. During various times in the 1970s the computer on which Macsyma ran was one of the most popular nodes on the ARPANET. We discuss the attempts in the late 70s and the 80s to develop Macsyma systems that ran on popular computer architectures. Finally, we discuss the impact of the fundamental ideas in Macsyma on the author’s current research on large scale engineering and socio-technical systems

On the factorization of polynomials over algebraic fields

SIGLEAvailable from British Library Document Supply Centre- DSC:DX86869 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Fast Computation of Special Resultants

We propose fast algorithms for computing composed products and composed sums, as well as diamond products of univariate polynomials. These operations correspond to special multivariate resultants, that we compute using power sums of roots of polynomials, by means of their generating series

On methods of computing galois groups and their implementations in MAPLE.

by Tang Ko Cheung, Simon.Thesis date on t.p. originally printed as 1997, of which 7 has been overwritten as 8 to become 1998.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 95-97).Chapter 1 --- Introduction --- p.5Chapter 1.1 --- Motivation --- p.5Chapter 1.1.1 --- Calculation of the Galois group --- p.5Chapter 1.1.2 --- Factorization of polynomials in a finite number of steps IS feasible --- p.6Chapter 1.2 --- Table & Diagram of Transitive Groups up to Degree 7 --- p.8Chapter 1.3 --- Background and Notation --- p.13Chapter 1.4 --- Content and Contribution of THIS thesis --- p.17Chapter 2 --- Stauduhar's Method --- p.20Chapter 2.1 --- Overview & Restrictions --- p.20Chapter 2.2 --- Representation of the Galois Group --- p.21Chapter 2.3 --- Groups and Functions --- p.22Chapter 2.4 --- Relative Resolvents --- p.24Chapter 2.4.1 --- Computing Resolvents Numerically --- p.24Chapter 2.4.2 --- Integer Roots of Resolvent Polynomials --- p.25Chapter 2.5 --- The Determination of Galois Groups --- p.26Chapter 2.5.1 --- Searching Procedures --- p.26Chapter 2.5.2 --- "Data: T(x1,x2 ,... ,xn), Coset Rcpresentatives & Searching Diagram" --- p.27Chapter 2.5.3 --- Examples --- p.32Chapter 2.6 --- Quadratic Factors of Resolvents --- p.35Chapter 2.7 --- Comment --- p.35Chapter 3 --- Factoring Polynomials Quickly --- p.37Chapter 3.1 --- History --- p.37Chapter 3.1.1 --- From Feasibility to Fast Algorithms --- p.37Chapter 3.1.2 --- Implementations on Computer Algebra Systems --- p.42Chapter 3.2 --- Squarefree factorization --- p.44Chapter 3.3 --- Factorization over finite fields --- p.47Chapter 3.4 --- Factorization over the integers --- p.50Chapter 3.5 --- Factorization over algebraic extension fields --- p.55Chapter 3.5.1 --- Reduction of the problem to the ground field --- p.55Chapter 3.5.2 --- Computation of primitive elements for multiple field extensions --- p.58Chapter 4 --- Soicher-McKay's Method --- p.60Chapter 4.1 --- "Overview, Restrictions and Background" --- p.60Chapter 4.2 --- Determining cycle types in GalQ(f) --- p.62Chapter 4.3 --- Absolute Resolvents --- p.64Chapter 4.3.1 --- Construction of resolvent --- p.64Chapter 4.3.2 --- Complete Factorization of Resolvent --- p.65Chapter 4.4 --- Linear Resolvent Polynomials --- p.67Chapter 4.4.1 --- r-sets and r-sequences --- p.67Chapter 4.4.2 --- Data: Orbit-length Partitions --- p.68Chapter 4.4.3 --- Constructing Linear Resolvents Symbolically --- p.70Chapter 4.4.4 --- Examples --- p.72Chapter 4.5 --- Further techniques --- p.72Chapter 4.5.1 --- Quadratic Resolvents --- p.73Chapter 4.5.2 --- Factorization over Q(diac(f)) --- p.73Chapter 4.6 --- Application to the Inverse Galois Problem --- p.74Chapter 4.7 --- Comment --- p.77Chapter A --- Demonstration of the MAPLE program --- p.78Chapter B --- Avenues for Further Exploration --- p.84Chapter B.1 --- Computational Galois Theory --- p.84Chapter B.2 --- Notes on SAC´ؤSymbolic and Algebraic Computation --- p.88Bibliography --- p.9