Multi-Valued Computer Algebra
Résumé
One of the main strengths of computer algebra is being able to solve a family of problems with one computation. In order to express not only one problem but a family of problems, one introduces some symbols which are in fact the parameters common to all the problems of the family. The user must be able to understand in which way these parameters affect the result when he looks at the answer. Otherwise it may lead to completely wrong calculations, which when used for numerical applications bring nonsensical answers. This is the case in most current Computer Algebra Systems we know because the form of the answer is never explicitly conditioned by the values of the parameters. The user is not even informed that the given answer may be wrong in some cases then computer algebra systems can not be entirely trustworthy. We have introduced multi-valued expressions called conditional expressions, in which each potential value is associated with a condition on some parameters. This is used, in particular, to capture the situation in integration, where the form of the answer can depend on whether certain quantities are positive, negative or zero. We show that it is also necessary when solving modular linear equations or deducing congruence conditions from complex expressions.