# Introduction

*Involutive bases*
[Gerdt 2005,
Gerdt, Blinkov 1998,
Apel 1998]
are particular generating sets for
ideals in polynomial rings, or more generally for submodules
of free modules over polynomial rings. Using an involutive
basis of the ideal under consideration allows e.g. to decide
easily whether a given element of the polynomial ring is
an element of the ideal or not.
More generally, involutive bases allow to compute in the
residue class ring modulo the given ideal.
Involutive bases form a special kind of Gröbner basis
[Buchberger 1965]. In
comparison to Gröbner bases, more combinatorial information
about the ideal or the residue class ring are
incorporated in the involutive basis; the algorithms computing
involutive bases are clearer from a structural point of view
because reduction of ideal or module generators modulo an
involutive basis is possible in a unique way only. During the
last ten years, V. P. Gerdt and Y. A. Blinkov have been
developing very efficient algorithms for the construction of
involutive bases [Gerdt 2005].
As a special case, *Janet bases*
[Janet 1929,
Plesken, Robertz 2005]
are involutive bases with respect to
Janet division.
Maurice Janet developed this kind of basis for the algebraic
analysis of (linear) systems of partial differential equations.
After his work seemed to be forgotten for several decades, J.-F.
Pommaret [Pommaret 1994]
pointed out that Janet's algorithm when applied to
linear PDEs with constant coefficients is a variant of
Buchberger's algorithm.

These web pages try to summarize some work on
involutive bases leading to implementations in Maple,
C++, and Python. In particular, involutive bases can be
computed for

- systems of commutative polynomials

(see *Involutive*
and *ginv*),

- linear systems of partial differential equations

(see *Janet*),

- systems of polynomials in certain Ore algebras,
(see
*JanetOre*),

- linear systems of difference equations
(see
*LDA*).

In the near future, the open source project
*ginv* will be able to
compute involutive bases in all of the above mentioned
contexts.

For more information about involutive bases, cf. also
http://invo.jinr.ru.