# Singular

##### 7.7.22.0. annRatSyz
Procedure from library `dmodloc.lib` (see dmodloc_lib).

Usage:
annRatSyz(f,g[,db,eng]); f, g polynomials, db,eng optional integers

Assume:
The basering is commutative and over a field of characteristic 0.

Return:
ring (a Weyl algebra) containing an ideal `LD', which is (part of) the annihilator of the rational function g/f in the corresponding Weyl algebra

Remarks:
This procedure uses the computation of certain syzygies. One can obtain the full annihilator by computing the Weyl closure of the ideal LD.

Note:
Activate the output ring with the `setring` command. In the output ring, the ideal `LD' (in Groebner basis) is (part of) the annihilator of g/f.
If db>0 is given, operators of order up to db are considered, otherwise, and by default, a minimal holonomic solution is computed.
If eng<>0, `std` is used for Groebner basis computations, otherwise, and by default, `slimgb` is used.
If printlevel =1, progress debug messages will be printed, if printlevel>=2, all the debug messages will be printed.

Example:
 ```LIB "dmodloc.lib"; // printlevel = 3; ring r = 0,(x,y),dp; poly f = 2*x*y; poly g = x^2 - y^3; def A = annRatSyz(f,g); // compute a holonomic solution setring A; A; ==> // coefficients: QQ ==> // number of vars : 4 ==> // block 1 : ordering dp ==> // : names x y Dx Dy ==> // block 2 : ordering C ==> // noncommutative relations: ==> // Dxx=x*Dx+1 ==> // Dyy=y*Dy+1 LD; ==> LD[1]=3*x*Dx+2*y*Dy+1 ==> LD[2]=y^4*Dy-x^2*y*Dy+2*y^3+x^2 setring r; def B = annRatSyz(f,g,5); // compute a solution up to degree 5 setring B; LD; // this is the full annihilator as we will check below ==> LD[1]=29223*y^2*Dx^2*Dy+2286*x*Dx*Dy^2-11464*y*Dy^3+87669*y*Dx^2+10304*Dy\ ^2 ==> LD[2]=111*y^3*Dx^2+2760*x^2*Dx^2+1470*x*y*Dx*Dy-296*y^2*Dy^2+4050*x*Dx ==> LD[3]=3*x*Dx+2*y*Dy+1 ==> LD[4]=y^3*Dy^2-x^2*Dy^2+6*y^2*Dy+6*y ==> LD[5]=y^4*Dy-x^2*y*Dy+2*y^3+x^2 setring r; def C = annRat(f,g); setring C; LD; // the full annihilator ==> LD[1]=3*y^2*Dx^2*Dy+2*x*Dx*Dy^2+9*y*Dx^2+4*Dy^2 ==> LD[2]=3*y^3*Dx^2-10*x*y*Dx*Dy-8*y^2*Dy^2+10*x*Dx ==> LD[3]=y^3*Dy^2-x^2*Dy^2-6*x*y*Dx+2*y^2*Dy+4*y ==> LD[4]=3*x*Dx+2*y*Dy+1 ==> LD[5]=y^4*Dy-x^2*y*Dy+2*y^3+x^2 ideal BLD = imap(B,LD); NF(LD,std(BLD)); ==> _[1]=0 ==> _[2]=0 ==> _[3]=0 ==> _[4]=0 ==> _[5]=0 ```