The Algorithmization of Physics: Math Between Science and Engineering - Artificial Intelligence and Symbolic Computation - page 6

I have proved that the 36 polynomials on the left-hand side of the respective
identities actually constitute a non-commutative Grobner basis with terminating
reduction (a fact that is not guaranteed in the noncommutative case, in partic-
ular not if one deals with infinitely many polynomials like the multiplication
operator

,theresult-
ing polynomial has just to be normalized with respect to this Grobner basis; the
corresponding normal form turns out to be the classical Green's operator with
the Green's function as its kernel - a well-known picture for every physicist!
As a conclusion, let me briefly reflect on the solution scheme sketched above.
First of all, we observe the algebraization involved in transforming the topological
concepts of differentiation and integration into fitting algebraic counterparts; the
creation of ￿

f above). Since I have retrieved a generic formula for G

can also be seen as an instance
of math-internal modeling in the sense described in the first section. Second we
note the crucial role of noncommutative Grobner bases in providing the classical
Green's operator

{D, A, B, L, R}∪{f|f ∈ F

#

}

: without a confluent and noetherian system of identities the
search for normal forms would not be an algorithmic process. Third we may also
notice the advantage of working purely on the level of operators, as opposed to
traditional solution methods on the function level that use costly determinant
computations; see e.g. page 189 in [11].
Finally, let me point out that solving BVPs in this way could be the first
step into a new territory for symbolic computation that we have baptized “sym-
bolic functional analysis” in [17]. Its common feature would be the algorithmic
study (and inversion) of some crucial operators occurring in functional analy-
sis; its main tool would be noncommutative polynomials. Besides more general
BVPs (PDEs rather than ODEs, nonlinear rather than linear, systems of equa-
tions rather than single equations, etc), some potentially interesting problems
could be: single layer potentials, exterior Dirichlet problems, inverse problems
like the backwards heat equation, computation of principal symbols, eigenvalue
problems.

G

References

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of Zero-Dimensional Polynomial Ideal (in German). PhD Thesis, University of
Innsbruck, Institute for Mathematics (1965)
2. Buchberger, B.: An Algorithmic Criterion for the Solvability of Algebraic Systems
of Equations (in German). Æquationes Mathematicae, 4 (1970) 374-383, in [8] 535-
545
3. Buchberger, B.: Editorial. Journal of Symbolic Computation 1/1 (1985)
4. Buchberger, B.: Logic for Computer Science. Lecture Notes, Johannes Kepler Uni-
versity, Linz, Austria (1991)
5. Buchberger, B.: Symbolic Computation (in German). In: Gustav Pomberger and
Peter Rechenberg, Handbuch der Informatik, Birkhauser, Munchen (1997) 955-974
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putation. In: Bruno Buchberger and George Edwin Collins and Rudiger Loos, Al-
gebraic Simplification, Springer, Wien (1982)