In this survey I could only touch on a few topics where computer algebra has contributed to mathematical research. Many others have not been mentioned, although there exist powerful algorithms and efficient implementations. In the first place, the computation of invariant rings for group actions of finite [Sturmfels1993,Kemper1996,Decker and De Jong1998], reductive [Derksen1997] or some uni-potent [Greuel, Pfister and Schönemann1990-1998] groups belong here. Computation of invariants have important applications for explicit construction of moduli spaces, for example, for vector bundles or for singularities [Frühbis-Krüger2000,T. Bayer2000] but also for dynamical systems with symmetries [Gatermann1999]. Libraries for computing invariants are available in SINGULAR. Available is also the Puiseux expansion (even better, the Hamburger-Noether expansion, cf. Lam for description of an implementation) of plane curve singularities. The latter is one of the few examples of an algorithm in algebraic geometry where Gröbner bases are not needed.
The applications of computer algebra and, in particular, of Gröbner bases in projective algebraic geometry are so numerous that I can only refer to the textbooks of Cox-Little-O'Shea, Eisenbud and Vasconcelos and the literature cited there. The applications include classification of varieties and vector bundles, cohomology, moduli spaces and fascinating problems in enumerative geometry.
However, there are also some important problems for which an algorithm is either not known or not yet implemented (for further open problems see also Ei1):
This is one of the most important tools for treating singular varieties. At least three approaches seem to be possible:
For surfaces we have Zariski's method of successive normalisation and blowing up points and the Hirzebruch-Jung method of resolving the discriminant curve of a projection.
For arbitrary varieties, new methods of Bierstone, Milman and Villamajor provide a constructive approach to resolution in the spirit of Hironaka. First attempts in this direction have been made by Schicho.
This is a little vague since I do not mean to actually compute with infinite power series, the input should be polynomials. However, it would be highly desirable to make effective use of the Weierstraß preparation theorem. This is related to the problem of elimination in power series rings.
Moreover, no algorithm seems to be known to compute an algebraic representative of the semi-universal deformation of an isolated singularity (which is known to exist).
Also, I do not know any algorithm for Hensel's lemma.
In this category falls, at least principally, the study of Gröbner bases over rings. This has, of course, been studied, cf. AL,Ka1, but I still consider the dependence of Gröbner bases on parameters as an unsolved problem (in the sense of an intrinsic or predictable description, if it exists).
In many cases, one is interested in finding equations for parameters describing precisely the locus where certain invariants jump. This is related to the above problem since Gröbner bases usually only give a sufficient but not necessary answer.
Mainly in practical applications of Gröbner bases to ``symbolic solving'', parameters are real or complex numbers. It would then be important to know, for which range of the parameters the symbolic solution holds.
The big success of numerical computations in real life problems seems to show that symbolic computation is of little use for such problems. However, as is well-known, symbolic preprocessing of a system of polynomial (even ordinary and partial differential) equations may not only lead to much better conditions for the system to be solved numerically but even make numerical solving possible.
There is continuous progress in this direction, cf. CLO2,Moe,Ve, not only by Gröbner basis methods. A completely different approach via multivariate resultants (cf. CE) has become favourable to several people due to the new sparse resultants by Ge. However, an implementation in SINGULAR (cf. We,Hil) does not show superiority of resultant methods, at least for many variables against triangular set methods of either Lazard or Möller. Nevertheless, much more has to be done.
The main disadvantage of symbolic methods in practical, real life applications is its complexity. Even if a system is able to return a symbolic answer in a short time, this answer is often not humanly interpretable. Therefore, a symbolic simplification is necessary, either before, during, or after generation. Of course, the result must still be approximately correct.
This leads to the problem of validity of ``simplified'' symbolic computation. A completely open subproblem is the validity resp. error estimation of Gröbner basis computations with floating point coefficients.
The simplification problem means providing simple and humanly understandable symbolic solutions which are approximately correct for numerical values in a region which can be specified. This problem belongs, in my opinion, perhaps to the most important ones in connection with applications of computer algebra to industrial and economical problems.
Before Gröbner bases were introduced by Buchberger, the so-called Ritt-Wu method, cf. Ri, was developed for symbolic computation in non-commutative rings of differential operators. However, nowadays, commutative Gröbner bases are implemented in almost every major computer algebra system, whilst only few systems provide non-commutative algorithms. Standard bases for some non-commutative structures have been implemented in the system FELIX [Apel and Klaus1991] as well as in an experimental version in SINGULAR; the system Bergman and an extension called Anick can compute Gröbner bases and higher syzygies in the non-commutative case.
Highly desirable are effective implementations for non-commutative Gröbner bases in the Weyl algebra, the Grassmannian, for D-modules or the enveloping algebra of a finite dimensional Lie algebra (the general theory being basically understood, cf. Mo2,Uv,Ap).
The recent textbook of SST shows a
wide variety of algorithms for modules over the Weyl algebra and
-modules for which an efficient implementation is missing.
But even classical algebraic geometry, as was shown, for example, by KM, has a natural embedding into non-commutative algebraic geometry. A special case is known as quantisation, a kind of non-commutative deformation of a commutative algebra.
Providing algorithms and implementations for the use of computer algebra in non-commutative algebraic geometry could become a task and challenge for a new generation of computer algebra systems.