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 PDEs a ranking of its derivatives must be provided; then the corresponding Janet basis is unique. If a system of linear PDEs 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 PDEs its Janet basis must be determined first.
Generating a Janet basis
Any system of linear homogeneous PDEs 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 PDEs in any number of dependent and independent variables and generated a normal form for them. Here mainly linear PDEs 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 PDEs, at first a ranking of its derivatives must be defined.
Definition: A ranking of derivatives is a total ordering such that for any two derivatives , and , and any derivation operator the relations and are valid.
A derivative is called higher than if . The highest
derivative in an equation is called its leading derivative. For the derivatives up to order two of a single function depending on and with two possible order are the LEX order and the GRLEX order .
Here the usual notation is used. If the number of functions is higher than one, these orderings have to be generalized appropriately, e.g. the orderings or may be applied. The first basic operation to be applied in generating a Janet basis is the reduction of an equation w.r.t. another one . In colloquial terms this means the following: Whenever a derivative of may be obtained from the leading derivative of by suitable differentiation, this differentiation is performed and the result is subtracted from . Reduction w.r.t. a system of PDEs means reduction w.r.t. all elements of the system. A system of linear PDEs is called autoreduced if all possible reductions have been performed.
The second basic operation for generating a Janet basis is the inclusion of integrability conditions. They are obtained as follows: If two equations and are such that by suitable differentiations two new equations may be obtained with like leading derivatives, by cross-multiplication with its leading coefficients and subtraction of the resulting equations a new equation is obtained, it is called an integrability condition. If by reduction w.r.t. the remaining equations of the system it does not vanish it is included as a new equation to the system.
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 , the Janet basis corresponding to is returned.
S1: (Autoreduction) Assign
S2: (Completion) Assign
S3: (Integrability conditions) Find all pairs of leading terms of and of such that differentiation w.r.t. a nonmultiplier and multipliers leads to and determine the integrability conditions S4: (Reduction of integrability conditions). For all assign S5: (Termination?) If all are zero return , otherwise make the assignment , reorder properly and goto S1