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- sm1.syz([f,v,w]|proc=p)
-
:: computes the syzygy of f in the ring of differential
operators with the variable v.
- return
-
List
- p
-
Number
- f, v, w
-
List
-
The return values is of the form
[s,[g, m, t]].
Here s is the syzygy of f in the ring of differential
operators with the variable v.
g is a Groebner basis of f with the weight vector w,
and m is a matrix that translates the input matrix f to the Gr\"obner
basis g.
t is the syzygy of the Gr\"obner basis g.
In summary, g = m f and
s f = 0 hold as matrices.
-
The weight vectors are given by w, which can be omitted.
If w is not given,
the graded reverse lexicographic order will be used to compute Grobner basis.
-
When a non-term order is given, the Grobner basis is computed in
the homogenized Weyl algebra (See Section 1.2 of the book of SST).
The homogenization variable h is automatically added.
[293] sm1.syz([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
[[[y*x*dy*dx-2,-x*dx-y*dy+1]], generators of the syzygy
[[[x*dx+y*dy-1],[y^2*dy^2+2]], grobner basis
[[1,0],[y*dy,-1]], transformation matrix
[[y*x*dy*dx-2,-x*dx-y*dy+1]]]]
[294]sm1.syz([[x^2*dx^2+x*dx+y^2*dy^2+y*dy-4,x*y*dx*dy-1],[x,y],[[dx,-1,x,1]]]);
[[[y*x*dy*dx-1,-x^2*dx^2-x*dx-y^2*dy^2-y*dy+4]], generators of the syzygy
[[[x^2*dx^2+h^2*x*dx+y^2*dy^2+h^2*y*dy-4*h^4],[y*x*dy*dx-h^4], GB
[h^4*x*dx+y^3*dy^3+3*h^2*y^2*dy^2-3*h^4*y*dy]],
[[1,0],[0,1],[y*dy,-x*dx]], transformation matrix
[[y*x*dy*dx-h^4,-x^2*dx^2-h^2*x*dx-y^2*dy^2-h^2*y*dy+4*h^4]]]]
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