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.
[1] It was first called the Janet basis by Fritz Schwarz in 1998.
[2] 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[3] 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.
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.
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.
Most results described here may be generalized in an obvious way to any number of variables or functions.
[4][5][6] In order to generate a unique representation for a given system of linear PDEs, at first a ranking of its derivatives must be defined.
[7] The first basic operation to be applied in generating a Janet basis is the reduction of an equation
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.
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.
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
is a subalgorithm that returns its argument with all possible reductions performed,
adds certain equations to the system in order to facilitate determining the integrability conditions.
To this end the variables are divides into multipliers and non-multipliers; details may be found in the above references.
Upon successful termination a Janet basis for the input system will be returned.
, i.e. the Janet basis for the originally given system is
The system is already autoreduced, i.e. step S1 returns it unchanged.
After a few more iterations finally the Janet basis
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 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.
[5] Janet's algorithm has been implemented in Maple.