Constructive methods in algebra and algebraic geometry are gaining more and more importance with the availability of computers and computer algebra softwares. Since its first version (see ), CASA has been designed to perform computations and utilize reasoning about algebraic and geometrical objects in classical affine and projective spaces over algebraically closed fields of characteristic zero. More precisely, the field has to be a computable field in the sense of the underlying computer algebra system Maple, i.e. all the arithmetic operations have to be available in the system. Usually, the field of computation is the rational numbers Q or a finite algebraic extension thereof.
Recently, new coding theoretic algorithms have been included. They require computations over finite fields.
Several people have contributed to the system in one way or another (see , , , , , , , , , , , , , , , , , , , , , , , , ).
The names of the main contributors are Klaus Aigner, Harald Deischinger, Ruediger Gebauer, Hong Gu, Michaela Hackl, Ralf Hemmecke (current coordinator), Erik Hillgarter, Michael Kalkbrener, Michal Mnuk, J. Rafael Sendra, Peter Stadelmeyer, Quoc-Nam Tran, Bernhard Wall, Geert Van de Weyer, Franz Winkler (director).
In the system, an algebraic set (a central notion in algebraic geometry) can be represented in four different ways:
The system provides a variety of operations on algebraic sets. As the efficiency of these operations is tightly bound to the way algebraic sets are represented, conversion routines are provided to support various views on one object, to deepen the understanding of its principles, and to speed up algorithms working on algebraic sets. CASA also works with the polynomial ideals corresponding to these geometric objects.
The basic operations available in CASA include:
Besides these basic operations, the following more advanced operations are available:
The major goal of CASA is to provide a comfortable, easy to use, efficient, flexible, and mathematically exact working environment for computational algebra and constructive algebraic geometry where all basic theoretical concepts map easily to available data structures.
CASA is built on the kernel of Maple and is fully independent of the operating system; hence, it can be used on every hardware where Maple is running.