2.1 A list of functions

2.1.1 Res_solv

Res_solv(m,d)

:: Find a solution u of the linear indefinite equation u m =d.

Res_solv(m,d,r)

:: Find a solution u of the linear indefinite equation u m =d. r is a ring object.

return

When [c,r] is the return value, c/r is the solution u.

m

Matrix or vector

d

Vector or scalar

• Find a solution u of the linear indefinite equation u m =d.
• It solves the linear indefinite equation in the ring of differential operators (with graded reverse lexicographic order) of the same set of variables of the ring to which m or d belongs. When the ring r is given, it solves the linear indefinite equation in the ring of differential operators (with graded reverse lexicographic order) of the same set of variables of the ring r.
• When m and d consist of constants, a ring r should be given.

In(16)= RingD("x,y");
In(17)= mm=[Dx,Dy,x];
In(18)= Res_solv(mm,1):
[    [    x , 0 , -Dx ]  , -1 ] 

The output means that -x*Dx + 0*Dy+Dx*x = 1.

In(4)=RingD("x");
     m=[ [x*Dx+2, 0],[Dx+3,x^3],[3,x],[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0]];
     d=[1,0];
     Res_solv(m,d):

[    [    x^2*Dx-x*Dx-4*x-1 , 0 , 0 , x ]  , -2 ] 

The output implies that -(1/2)*(x^2*Dx-x*Dx-4*x-1)*[x*Dx+2, 0]-(1/2)*[Dx*(x*Dx+3)-(x*Dx+2)*(x*Dx-4),0] = [1,0]

In(4)= r=RingD("x,y");
In(5)= Res_solv([[1,2],[3,4]],[5,0],r):
[    [    10 , -5 ]  , -1 ] 

See also

Res_solv_h, Kernel, GetRing, SetRing.

Files

lib/restriction/complex.k

2.1.2 Res_solv2

Res_solv2(m,v,j)

:: Find a solution u of the linear indefinite equation u m =v mod j.

Res_solv2(m,v,j,r)

:: Find a solution u of the linear indefinite equation u m =v mod j. r is a ring object.

return

When [c,r] is the return value, c/r is the solution u.

m

Matrix or vector

v j

Vector or scalar

• Find a solution u of the linear indefinite equation u m =v mod j.
• Let m be the left D-homomorphism m : D^p —> D^q/j. The function returns an element in m^(-1)(v).
• It solves the linear indefinite equation in the ring of differential operators (with graded reverse lexicographic order) of the same set of variables of the ring to which m or v belongs. When the ring r is given, it solves the linear indefinite equation in the ring of differential operators (with graded reverse lexicographic order) of the same set of variables of the ring r.
• When m and v consist of constants, a ring r should be given.

In(28)= r=RingD("x,y");
In(29)= Res_solv2([x,y],[x^2+y^2],[x]):
[    [    0 , y ]  , 1 ] 

The output means that 0*x + y*y = x^2+y^2 mod x

In(32)= Res_solv2([x,y],[x^2+y^2],[],r):
[    [    x , y ]  , 1 ] 

The output implies that x*x + y*y = x^2+y^2.

See also

Res_solv2_h, Kernel2, GetRing, SetRing.

Files

lib/restriction/complex.k

2.1.3 Kernel

Kernel(m)

:: Find solution basis of the linear indefinite equation u m =0.

Kernel(m,r)

:: Find solution basis of the linear indefinite equation u m =0. r is a ring object.

return

List

m

Matrix or vector

• Find solution basis of the linear indefinite equation u m =0.
• When the return value is k, k[0] is a set of generators of the kernel. k[1] is [gb, backward transformation, syzygy without dehomogenization].
• It finds the kernel in the ring to which m belongs. When the ring r is given, it finds the kernel in the ring r.
• When m consists of constants, a ring r should be given.

In(16)= RingD("x,y");
In(17)= mm=[[Dx],[Dy],[x]];
In(18)= Pmat(Kernel(mm));
 [ 
  [ 
    [    -x*Dx-2 , 0 , Dx^2 ] 
    [    -x*Dy , -1 , Dx*Dy ] 
    [    -x^2 , 0 , x*Dx-1 ] 
  ]
  [ 
   [ 
     [    -1 ] 
   ]
   [ 
     [    x , 0 , -Dx ] 
   ]
   [ 
     [    -x*Dx-2 , 0 , Dx^2 ] 
     [    -x*Dy , -1 , Dx*Dy ] 
     [    -x^2 , 0 , x*Dx-1 ] 
   ]
  ]
 ]

In(4)= r=RingD("x,y");
In(5)= k=Kernel([[1,2],[2,4]],r); k[0]:
[    [    2 , -1 ]  ] 

See also

Kernel_h, Res_solv, GetRing, SetRing.

Files

lib/restriction/complex.k

2.1.4 Kernel2

Kernel2(m)

:: Get the kernel of m : D^p —> D^q/j.

Kernel2(m,r)

:: Get the kernel of m : D^p —> D^q/j. r is a ring object.

return

List

m j

Matrix or vector

• Get a set of generators of the the kernel of m : D^p —> D^q/j.
• D^p is a set of row vectors. When u is an element of D^p, define the map from D^p to D^q/j by u m.
• It finds the kernel in the ring to which m belongs. When the ring r is given, it finds the kernel in the ring r.
• When m consists of constants, a ring r should be given.

In(27)= r=RingD("x,y");
In(28)= Kernel2([[x,y],[x^2,x*y]],[]):
[    [    -x , 1 ]  ] 
In(29)=Kernel2([[x,y],[x^2,x*y]],[[x,y]]):
[    [    1 , 0 ]  , [    0 , 1 ]  ] 

In(41)=Kernel2([0],[0],r):
[    [    1 ]  , [    0 ]  ] 
In(42)=Kernel2([[0,0],[0,0]],[[0,0]],r):
[    [    1 , 0 ]  , [    0 , 1 ]  , [    0 , 0 ]  ] 
In(43)=Kernel2([[0,0,0],[0,0,0]],[],r):
[    [    1 , 0 ]  , [    0 , 1 ]  ] 

See also

Kernel2_h, Res_solv2, GetRing, SetRing, Kernel

Files

lib/restriction/complex.k

2.1.5 Gb

Gb(f)

:: It computes the Grobner basis of f.

Gb(m,r)

:: It computes the Grobner basis of f. r is a ring object.

Gb_h(f)

:: It computes the Grobner basis of f.

Gb_h(m,r)

:: It computes the Grobner basis of f. r is a ring object.

return

List

f

Matrix or vector

• Compute the Grobner basis of f.
• Functions with _h computes Grobner bases in the homogenized Weyl algebra.
• When the return value is k, k[0] is the Grobner basis. k[1] is the initial ideal or the initial module of f, when the ring is defined with a weight vector.
• It computes the Grobner basis in the ring to which f belongs. When the ring r is given, it computes the Grobner basis in the ring r.
• When f consists of constants, a ring r should be given.

In(5)= r=RingD("x,y");
In(6)= m=[[x^2+y^2-1],[x*y-1]];
In(7)= Gb(m):
[    [    [    x^2+y^2-1 ]  , [    x*y-1 ]  , [    y^3+x-y ]  ]  , 
[    [    x^2+y^2-1 ]  , [    x*y-1 ]  , [    y^3+x-y ]  ]  ] 

In(11)= RingD("x,y",[["x",1]]);
In(12)= r=RingD("x,y",[["x",1]]);
In(13)= Gb(m,r):
[    [    [    x+y^3-y ]  , [    -y^4+y^2-1 ]  ]  , 
[    [    x ]  , [    -y^4+y^2-1 ]  ]  ] 

See also

Gb_h, Kernel, Res_solv, RingD, GetRing, SetRing.

Files

lib/restriction/complex.k

2.1.6 Res_shiftMatrix

Res_shiftMatrix(m,v)

:: Generate a matrix associated to a degree shift vector m

Res_shiftMatrix(m,v,r)

:: Generate a matrix associated to a degree shift vector m r is a ring object.

return

Matrix

m

Vector

v

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• Returns n by n matrix diag(v^(m1), ..., v^(mn))

In(5)= r=RingD("x,y");
In(6)= Res_shiftMatrix([-1,0,3],x):
[    [    x^(-1) , 0 , 0 ]  , [    0 , 1 , 0 ]  , [    0 , 0 , x^3 ]  ] 

In(9)=  rrr = RingD("t,x,y",[["t",1,"x",-1,"y",-1,"Dx",1,"Dy",1]]);
In(10)=  m=[Dx-(x*Dx+y*Dy+2),Dy-(x*Dx+y*Dy+2)];
In(12)=  m=Gb(m);
In(13)=  k = Kernel_h(m[0]);
In(14)=  Pmat(k[0]);
 [ 
   [    -Dy+3*h , Dx-3*h , 1 ] 
   [    -x*Dx+x*Dy-y*Dy-3*x*h , y*Dy+3*x*h , h-x ] 
 ]

In(15)=Pmat(m[0]);
  [    Dx*h-x*Dx-y*Dy-2*h^2 , Dy*h-x*Dx-y*Dy-2*h^2 , 
       x*Dx^2-x*Dx*Dy+y*Dx*Dy-y*Dy^2 ] 

In(18)=k2 = Gb_h(k[0]*Res_shiftMatrix([1,1,1],t));
In(19)=Pmat(Substitute(k2[0],t,1));
 [ 
   [    -Dy+3*h , Dx-3*h , 1 ] 
   [    -x*Dx+x*Dy-y*Dy-3*x*h , y*Dy+3*x*h , h-x ] 
 ]

See also

Gb, (m,(u,v))-Grobner basis

Files

lib/restriction/complex.k

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