<s>
In	O
mathematics	O
,	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
is	O
a	O
normal	O
form	O
for	O
systems	O
of	O
linear	O
homogeneous	O
partial	O
differential	O
equations	O
(	O
PDEs	O
)	O
that	O
removes	O
the	O
inherent	O
arbitrariness	O
of	O
any	O
such	O
system	O
.	O
</s>
<s>
It	O
was	O
first	O
called	O
the	O
Janet	B-Algorithm
basis	I-Algorithm
by	O
Fritz	O
Schwarz	O
in	O
1998	O
.	O
</s>
<s>
A	O
Janet	B-Algorithm
basis	I-Algorithm
is	O
the	O
predecessor	O
of	O
a	O
Gröbner	O
basis	O
introduced	O
by	O
Bruno	O
Buchberger	O
for	O
polynomial	O
ideals	O
.	O
</s>
<s>
In	O
order	O
to	O
generate	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
for	O
any	O
given	O
system	O
of	O
linear	O
PDEs	O
a	O
ranking	O
of	O
its	O
derivatives	O
must	O
be	O
provided	O
;	O
then	O
the	O
corresponding	O
Janet	B-Algorithm
basis	I-Algorithm
is	O
unique	O
.	O
</s>
<s>
If	O
a	O
system	O
of	O
linear	O
PDEs	O
is	O
given	O
in	O
terms	O
of	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
its	O
differential	O
dimension	O
may	O
easily	O
be	O
determined	O
;	O
it	O
is	O
a	O
measure	O
for	O
the	O
degree	O
of	O
indeterminacy	O
of	O
its	O
general	O
solution	O
.	O
</s>
<s>
In	O
order	O
to	O
generate	O
a	O
Loewy	O
decomposition	O
of	O
a	O
system	O
of	O
linear	O
PDEs	O
its	O
Janet	B-Algorithm
basis	I-Algorithm
must	O
be	O
determined	O
first	O
.	O
</s>
<s>
The	O
first	O
basic	O
operation	O
to	O
be	O
applied	O
in	O
generating	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
is	O
the	O
reduction	O
of	O
an	O
equation	O
w.r.t.	O
</s>
<s>
The	O
second	O
basic	O
operation	O
for	O
generating	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
is	O
the	O
inclusion	O
of	O
integrability	O
conditions	O
.	O
</s>
<s>
It	O
may	O
be	O
shown	O
that	O
repeating	O
these	O
operations	O
always	O
terminates	O
after	O
a	O
finite	O
number	O
of	O
steps	O
with	O
a	O
unique	O
answer	O
which	O
is	O
called	O
the	O
Janet	B-Algorithm
basis	I-Algorithm
for	O
the	O
input	O
system	O
.	O
</s>
<s>
Janet	O
's	O
algorithm	O
:	O
Given	O
a	O
system	O
of	O
linear	O
differential	O
polynomials	O
,	O
the	O
Janet	B-Algorithm
basis	I-Algorithm
corresponding	O
to	O
is	O
returned	O
.	O
</s>
<s>
Upon	O
successful	O
termination	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
for	O
the	O
input	O
system	O
will	O
be	O
returned	O
.	O
</s>
<s>
the	O
Janet	B-Algorithm
basis	I-Algorithm
for	O
the	O
originally	O
given	O
system	O
is	O
with	O
the	O
trivial	O
solution	O
.	O
</s>
<s>
The	O
most	O
important	O
application	O
of	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
is	O
its	O
use	O
for	O
deciding	O
the	O
degree	O
of	O
indeterminacy	O
of	O
a	O
system	O
of	O
linear	O
homogeneous	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
answer	O
may	O
be	O
more	O
involved	O
,	O
there	O
may	O
be	O
infinitely	O
many	O
free	O
constants	O
in	O
the	O
general	O
solution	O
;	O
they	O
may	O
be	O
obtained	O
from	O
the	O
Loewy	O
decomposition	O
of	O
the	O
respective	O
Janet	B-Algorithm
basis	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
the	O
Janet	B-Algorithm
basis	I-Algorithm
of	O
a	O
module	O
allows	O
to	O
read	O
off	O
a	O
Janet	B-Algorithm
basis	I-Algorithm
for	O
the	O
syzygy	O
module	O
.	O
</s>
