sm1_gb

sm1_gb([f,v,w]|proc=p,sorted=q)

:: computes the Grobner basis of f in the ring of differential operators with the variable v.

sm1_gb_d([f,v,w]|proc=p)

:: computes the Grobner basis of f in the ring of differential operators with the variable v. The result will be returned as a list of distributed polynomials.

return

List

p, q

Number

f, v, w

List

• It returns the Grobner basis of the set of polynomials f in the ring of deferential operators with the variables v.
• 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.
• The return value of sm1_gb is the list of the Grobner basis of f and the initial terms (when w is not given) or initial ideal (when w is given).
• sm1_gb_d returns the results by a list of distributed polynomials. Monomials in each distributed polynomial are ordered in the given order. The return value consists of [variable names, order matrix, grobner basis in districuted polynomials, initial monomials or initial polynomials].
• 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.
• When the optional variable q is set, sm1_gb returns, as the third return value, a list of the Grobner basis and the initial ideal with sums of monomials sorted by the given order. Each polynomial is expressed as a string temporally for now.

[293] sm1_gb([[x*dx+y*dy-1,x*y*dx*dy-2],[x,y]]);
[[x*dx+y*dy-1,y^2*dy^2+2],[x*dx,y^2*dy^2]]

In the example above,

[294] sm1_gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]);
[[dx+dy^3-4*dy,-dy^4+4*dy^2-1],[dx,-dy^4]]

In the example above, two monomials

[294] F=sm1_gb([[dx^2+dy^2-4,dx*dy-1],[x,y],[[dx,50,dy,2,x,1]]]|sorted=1);
      map(print,F[2][0])$
      map(print,F[2][1])$

[595]
   sm1_gb([["dx*(x*dx +y*dy-2)-1","dy*(x*dx + y*dy -2)-1"],
             [x,y],[[dx,1,x,-1],[dy,1]]]);

[[x*dx^2+(y*dy-h^2)*dx-h^3,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx-h^3*dy],
 [x*dx^2+(y*dy-h^2)*dx,x*dy*dx+y*dy^2-h^2*dy-h^3,h^3*dx]]

[596]
   sm1_gb_d([["dx (x dx +y dy-2)-1","dy (x dx + y dy -2)-1"],
             "x,y",[[dx,1,x,-1],[dy,1]]]);
[[[e0,x,y,H,E,dx,dy,h],
 [[0,-1,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],
  [0,1,1,1,1,1,1,0],[0,0,0,0,0,0,-1,0],[0,0,0,0,0,-1,0,0],
  [0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0],[0,0,-1,0,0,0,0,0],
  [0,0,0,0,0,0,0,1]]],
[[(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>+(-1)*
<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0
,0,0,0,1,2>>+(-1)*<<0,0,0,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>+(-1)*<<0,0,0,0,0,0
,1,3>>],
 [(1)*<<0,0,1,0,0,1,1,0>>+(1)*<<0,1,0,0,0,2,0,0>>+(-1)*<<0,0,0,0,0,1,0,2>>,(1)*<
<0,0,1,0,0,0,2,0>>+(1)*<<0,1,0,0,0,1,1,0>>+(-1)*<<0,0,0,0,0,0,1,2>>+(-1)*<<0,0,0
,0,0,0,0,3>>,(1)*<<0,0,0,0,0,1,0,3>>]]]

Reference

sm1_reduction, sm1_rat_to_p

• sm1_deRham

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