ComputeralgebraSystemen

Computeralgebra, also known as computer algebra or symbolic computation, is a field at the intersection of algebra and computer science that studies algorithms for manipulating mathematical expressions exactly and for solving mathematical problems symbolically. It focuses on representing and transforming objects such as polynomials, rational functions, algebraic numbers, and special functions, without recourse to floating point approximations. Typical tasks include simplification, expansion, factorization, solving algebraic equations, symbolic differentiation and integration, and the manipulation of series and differential equations.

One cornerstone of the discipline is Gröbner bases, introduced by Buchberger in 1965, which provides systematic methods for solving systems of multivariate polynomial equations, performing variable elimination, and computing algebraic invariants. Other important techniques include polynomial factorization, gcd computations, partial fraction decomposition, and algorithms for symbolic integration and solving differential equations. The field also encompasses algebraic geometry, number theory, and the design of data structures and algorithms for exact arithmetic over integers and rational numbers.

Computer algebra systems (CAS) integrate these algorithms into interactive software. Notable systems include Mathematica, Maple, Magma, Maxima, SageMath, and MuPAD (historical); several are open source while others are commercial. CAS platforms often combine exact symbolic manipulation with numeric evaluation, visualization, and scripting capabilities, and may interoperate with other mathematical software and libraries.

Applications span pure and applied mathematics, physics, engineering, computer science, cryptography, and education. They are used for research in symbolic proofs, model building, and automated reasoning, as well as for solving large-scale algebraic systems arising in engineering and geometry. Limitations include high computational complexity for general problems and expression swell, which can produce very large intermediate expressions. Ongoing research aims to improve efficiency and to integrate symbolic and numeric methods.