## Section: Overall Objectives

### Effective algebraic theories and their implementations in computer algebra systems

Although not yet very popular in applied mathematics, algebraic and differential geometric approaches of functional systems have lengthly been studied in foundamental mathematics. We can state a few names of mathematical theories such as (differential) Galois theory, Lie groups, exterior differential systems, differential algebra, algebraic analysis, etc. [36], [41], [51], [67], [75], [76], [81], [85], [101], [104], [109].

Over the past years, some of these algebraic theories for the study of functional systems have been investigated in the computer algebra community within an algorithmic viewpoint, mostly driven by applications to engineering sciences such as mathematical systems theory and control theory.

Gröbner or Janet bases [40], [102]
for noncommutative polynomial
rings of functional operators or differential elimination
techniques for differential
systems [45], [46], [72],
based on differential algebra [101], [76],
are remarkable examples of those effective algebraic methods. They
are nowadays implemented in standard computer algebra systems (e.g. `Maple` , `Mathematica` , `Magma` ).

These effective algebraic approaches also form the algorithmic “engines” at the basis of the first developments of effective versions of modern algebraic theories (algebraic geometry, differential algebra, module theory and homological algebra over certain noncommutative polynomial rings of functional operators, algebraic analysis, etc.).

The above-mentioned results are just the tip of the iceberg and much more effort must be made in the future for making effective larger parts of fundamental mathematics and making them largely available in standard computer algebra systems. This “democratization” process towards the accessibility of fundamental mathematics is important, for instance, for educational issues where computers can be used to teach them to students, scientists of other communities and engineerings, and for learning by doing and computing. Further developing effective mathematics, making them accessible to a larger audience through dedicated software, and demonstrating them through interesting engineering problems are at the core of the GAIA team. The latter engineering problems are major sources of motivation for the development of effective algebraic theories and their implementations in computer algebra systems as explained in the next section.