In mathematics, a Janet basis is a normal form for systems of linear homogeneous partial differential equations (PDEs) that removes the inherent arbitrariness of any such system. It was introduced in 1920 by Maurice Janet. It was first called the Janet basis by Fritz Schwarz in 1998.
Contents
The left hand sides of such systems of equations may be considered as differential polynomials of a ring, and Janet's normal form as a special basis of the ideal that they generate. By abuse of language, this terminology will be applied both to the original system and the ideal of differential polynomials generated by the left hand sides. A Janet basis is the predecessor of a Groebner basis introduced by Bruno Buchberger for polynomial ideals. In order to generate a Janet basis for any given system of linear pde's a ranking of its derivatives must be provided; then the corresponding Janet basis is unique. If a system of linear pde's is given in terms of a Janet basis its differential dimension may easily be determined; it is a measure for the degree of indeterminacy of its general solution. In order to generate a Loewy decomposition of a system of linear pde's its Janet basis must be determined first.
Generating a Janet basis
Any system of linear homogeneous pde's is highly non-unique, e.g. an arbitrary linear combination of its elements may be added to the system without changing its solution set. A priori it is not known whether it has any nontrivial solutions. More generally, the degree of arbitrariness of its general solution is not known, i.e. how many undetermined constants or functions it may contain. These questions were the starting point of Janet's work; he considered systems of linear pde's in any number of dependent and independent variables and generated a normal form for them. Here mainly linear pde's in the plane with the coordinates
Definition A ranking of derivatives is a total ordering such that for any two derivatives
A derivative
Here the usual notation
The second basic operation for generating a Janet basis is the inclusion of integrability conditions. They are obtained as follows: If two equations
It may be shown that repeating these operations always terminates after a finite number of steps with a unique answer which is called the Janet basis for the input system. Janet has organized them in terms of the following algorithm.
Janet's Algorithm Given a system of linear differential polynomials
and determine the integrability conditions
S4: (Reduction of integrability conditions). For allHere
Example 1 Let the system
Steps S3 and S4 generate the integrability condition
The next example involves two unknown functions
Example 2 Consider the system
in
Upon reduction in step S4 they are
In step S5 they are included into the system and the algorithms starts again with step S1 with the extended system. After a few more iterations finally the Janet basis
is obtained. It yields the general solution
Janet's algorithm has been implemented in Maple
Application of Janet bases
The most important application of a Janet basis is its use for deciding the degree of indeterminacy of a system of linear homogeneous partial differential equations. The answer in the above Example 1 is that the system under consideration allows only the trivial solution. In the second Example 2 a two-dimensional solution space is obtained. In general, the answer may be more involved, there may be infinitely many free constants in the general solution; they may be obtained from the Loewy decomposition of the respective Janet basis. Furthermore, the Janet basis of a module allows to read off a Janet basis for the syzygy module