Janet basis
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.
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 Gröbner 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.
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 and will be considered; the number of unknown functions is one or two. Most results described here may be generalized in an obvious way to any number of variables or functions. In order to generate a unique representation for a given system of linear pde's, at first a ranking of its derivatives must be defined.
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