Computation of annihilating ideals by kan/sm1 and ox_asir

Let $D = {\bf Q} \langle x_1, \ldots, x_n , \partial_{1} , \ldots, \partial_{n} \rangle$ be the ring of differential operators. For a given polynomial $f \in {\bf Q}[x_1, \ldots, x_n]$, the annihilating ideal of $f^{-1}$ is defined as

$${\rm Ann}\, f^{-1} = \{ \ell \in D \,\vert\, \ell \bullet f^{-1} = 0 \}.$$

Here, $\bullet$ denotes the action of $D$ to functions. The annihilating ideal can be regarded as the maximal differential equations for the function $f^{-1}$. 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 $D$ 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 $D$, 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 $f = x^3 - y^2 z^2$.

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 $-12(s+1)(3s+5)(3s+4)(6s+5)(6s+7)$ and the minimal integral root is $-1$ as shown in the output.

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