### Computation of annihilating ideals by kan/sm1 and ox_asir

Let be the ring of differential operators. For a given polynomial , the annihilating ideal of is defined as

Here, denotes the action of to functions. The annihilating ideal can be regarded as the maximal differential equations for the function . An algorithm to determine generators of the annihilating ideal was given by Oaku (see, e.g., [16, 5.3]). His algorithm reduces the problem to computations of Gröbner bases in and to find the minimal integral root of a polynomial. This algorithm (the function annfs) is implemented by kan/sm1 [19], for Gröbner basis computation in , and ox_asir, to factorize polynomials to find the integral roots. These two OpenXM compliant systems are integrated by the OpenXM protocol.

For example, the following is a sm1 session to find the annihilating ideal for .

sm1>[(x^3-y^2 z^2) (x,y,z)] annfs ::
Starting ox_asir server.
Byte order for control process is network byte order.
Byte order for engine process is network byte order.
[[-y*Dy+z*Dz, 2*x*Dx+3*y*Dy+6, -2*y*z^2*Dx-3*x^2*Dy,
-2*y^2*z*Dx-3*x^2*Dz, -2*z^3*Dx*Dz-3*x^2*Dy^2-2*z^2*Dx],
[-1,-139968*s^7-1119744*s^6-3802464*s^5-7107264*s^4
-7898796*s^3-5220720*s^2-1900500*s-294000]]

The last polynomial is factored as and the minimal integral root is as shown in the output.

Similarly, an algorithm to stratify singularity [13] is implemented by kan/sm1 [19], for Gröbner basis computation in , and ox_asir, for primary ideal decompositions.

Nobuki Takayama 2017-03-30