Basic Examples

Create a set of polynomials:
polys={x^2+3*y-2*z^4+7,5*x^2*y*z-4*y^2*z^2+3,x^2+y^2+z^3-6};
In[]:=
First compute its Groebner basis:
gb1=GroebnerBasis[polys,{x,y,z}]
In[]:=
{9-1890z+1205
2
z
+14820
3
z
+2794
4
z
-2756
5
z
+6198
6
z
-4256
7
z
-607
8
z
-130
9
z
-2324
10
z
+304
11
z
+64
12
z
+100
13
z
+200
14
z
,-174568722857469+32134376477706y+174412495017740z+1978174579982874
2
z
+338799254181097
3
z
-360676216949111
4
z
+836357857920969
5
z
-584758935888737
6
z
-68963401625500
7
z
-20615441826364
8
z
-309573453014396
9
z
+42730628534164
10
z
+7055457141820
11
z
+13177425497200
12
z
+26562952946000
13
z
,249548934638783+10711458825902
2
x
-174412495017740z-1978174579982874
2
z
-338799254181097
3
z
+339253299297307
4
z
-836357857920969
5
z
+584758935888737
6
z
+68963401625500
7
z
+20615441826364
8
z
+309573453014396
9
z
-42730628534164
10
z
-7055457141820
11
z
-13177425497200
12
z
-26562952946000
13
z
}
Out[]=
Compute the extended Groebner basis:
{gb2,cmat}=
[◼]
ExtendedGroebnerBasis
[polys,{x,y,z}]
In[]:=
{{9-1890z+1205
2
z
+14820
3
z
+2794
4
z
-2756
5
z
+6198
6
z
-4256
7
z
-607
8
z
-130
9
z
-2324
10
z
+304
11
z
+64
12
z
+100
13
z
+200
14
z
,-174568722857469+32134376477706y+174412495017740z+1978174579982874
2
z
+338799254181097
3
z
-360676216949111
4
z
+836357857920969
5
z
-584758935888737
6
z
-68963401625500
7
z
-20615441826364
8
z
-309573453014396
9
z
+42730628534164
10
z
+7055457141820
11
z
+13177425497200
12
z
+26562952946000
13
z
,249548934638783+10711458825902
2
x
-174412495017740z-1978174579982874
2
z
-338799254181097
3
z
+339253299297307
4
z
-836357857920969
5
z
+584758935888737
6
z
+68963401625500
7
z
+20615441826364
8
z
+309573453014396
9
z
-42730628534164
10
z
-7055457141820
11
z
-13177425497200
12
z
-26562952946000
13
z
},{{-45z-15yz+113
2
z
+975y
2
z
-400
2
y
2
z
+1140
3
z
-60y
3
z
-60
2
y
3
z
+208
4
z
-48y
4
z
-225
5
z
-75y
5
z
+580
6
z
-170y
6
z
+50
2
y
6
z
-136
7
z
-32
8
z
-100
10
z
,3-435z+80yz-88
2
z
+12y
2
z
+15
4
z
+64
5
z
-10y
5
z
+8
6
z
,45z-113
2
z
+1200y
2
z
-1140
3
z
+500y
3
z
-208
4
z
+48y
4
z
+225
5
z
-580
6
z
-150y
6
z
+136
7
z
-40y
7
z
+32
8
z
+100
10
z
},{15926908215185z+130275989040585yz-53557294129510
2
y
z+152243446089993
2
z
-10043547678483y
2
z
-7197744002550
2
y
2
z
+25622276783275
3
z
-5152326625065y
3
z
-348165578250
2
y
3
z
-28479789034230
4
z
-9872664025320y
4
z
-26012743950
2
y
4
z
+78251351237524
5
z
-22578510004100y
5
z
+6640738236500
2
y
5
z
-19588443609652
6
z
-3553741314860
7
z
+52025487900
8
z
-13281476473000
9
z
,-58189574285823+10711458825902y-10879103926555z+1439548800510yz-330073062345
2
z
+69633115650y
2
z
+1903218666174
3
z
+5202548790y
3
z
+8495982903688
4
z
-1328147647300y
4
z
+1062518117840
5
z
,-15926908215185z+160671882388530yz-152243446089993
2
z
+64439067311258y
2
z
-25622276783275
3
z
+6802691936790y
3
z
+28479789034230
4
z
+356570694450y
4
z
-78251351237524
5
z
-19901404514340y
5
z
+19588443609652
6
z
-5312590589200y
6
z
+3553741314860
7
z
-52025487900
8
z
+13281476473000
9
z
},{10711458825902-15926908215185z-130275989040585yz+53557294129510
2
y
z-152243446089993
2
z
+10043547678483y
2
z
+7197744002550
2
y
2
z
-25622276783275
3
z
+5152326625065y
3
z
+348165578250
2
y
3
z
+28479789034230
4
z
+9872664025320y
4
z
+26012743950
2
y
4
z
-78251351237524
5
z
+22578510004100y
5
z
-6640738236500
2
y
5
z
+19588443609652
6
z
+3553741314860
7
z
-52025487900
8
z
+13281476473000
9
z
,58189574285823-10711458825902y+10879103926555z-1439548800510yz+330073062345
2
z
-69633115650y
2
z
-1903218666174
3
z
-5202548790y
3
z
-8495982903688
4
z
+1328147647300y
4
z
-1062518117840
5
z
,15926908215185z-160671882388530yz+152243446089993
2
z
-64439067311258y
2
z
+25622276783275
3
z
-6802691936790y
3
z
-28479789034230
4
z
-356570694450y
4
z
+78251351237524
5
z
+19901404514340y
5
z
-19588443609652
6
z
+5312590589200y
6
z
-3553741314860
7
z
+52025487900
8
z
-13281476473000
9
z
}}}
Out[]=
Check that the bases agree up to a constant factor:
Together[gb1/gb2]
In[]:=
{1,1,1}
Out[]=
Check the conversion matrix identity:
Expand[cmat.polys-gb2]
In[]:=
{0,0,0}
Out[]=

Scope

Here is a larger example:
polys={x^2+3*y-2*z^4+7,-x^3*z+5*x^2*y^2*z-4*y^2*z^3+3,x^3+y^3+2*z^3-6};​​gb1=GroebnerBasis[polys,{x,y,z},MonomialOrderDegreeReverseLexicographic]
In[]:=
{-6+
3
x
+
3
y
+2
3
z
,-7-
2
x
-3y+2
4
z
,10+
2
x
+3y-6z+5
2
x
2
y
z+
3
y
z-4
2
y
3
z
,-6-10x-3xy+35
2
y
+5
2
x
2
y
+16
3
y
+6xz-30
2
y
z-x
3
y
z+5
5
y
z+2
3
z
+4x
2
y
3
z
,210+30
2
x
+90y+56
2
y
-150x
2
y
-167
2
x
2
y
-11
3
y
-80
2
x
3
y
-15
4
y
+25x
5
y
-40z-4
2
x
z-12yz+24
2
z
+140
2
y
2
z
+56
3
y
2
z
-100
3
z
-10
2
x
3
z
-30y
3
z
+50x
2
y
3
z
,35670-3200x-240
2
x
+31830y+300xy+150
2
x
y+25752
2
y
-31189
2
x
2
y
-7077
3
y
+750x
3
y
-27005
2
x
3
y
-5065
4
y
-6025
2
x
4
y
-8700
5
y
+625
8
y
+1070z-600xz+1032
2
x
z-304yz+260
2
x
yz-120
2
y
z-7000x
2
y
z-2900x
3
y
z-192
2
z
+3250x
2
z
-600
2
x
2
z
+240y
2
z
+750xy
2
z
+14630
2
y
2
z
+17502
3
y
2
z
+4760
4
y
2
z
-16900
3
z
+830
2
x
3
z
-13010y
3
z
-50
2
x
y
3
z
-2400
2
y
3
z
-800x
2
y
3
z
+250x
3
y
3
z
}
Out[]=
{gb2,cmat}=
[◼]
ExtendedGroebnerBasis
[polys,{x,y,z},MonomialOrderDegreeReverseLexicographic]
In[]:=
{{-6+
3
x
+
3
y
+2
3
z
,-7-
2
x
-3y+2
4
z
,10+
2
x
+3y-6z+5
2
x
2
y
z+
3
y
z-4
2
y
3
z
,-6-10x-3xy+35
2
y
+5
2
x
2
y
+16
3
y
+6xz-30
2
y
z-x
3
y
z+5
5
y
z+2
3
z
+4x
2
y
3
z
,210+30
2
x
+90y+56
2
y
-150x
2
y
-167
2
x
2
y
-11
3
y
-80
2
x
3
y
-15
4
y
+25x
5
y
-40z-4
2
x
z-12yz+24
2
z
+140
2
y
2
z
+56
3
y
2
z
-100
3
z
-10
2
x
3
z
-30y
3
z
+50x
2
y
3
z
,71340-6400x-480
2
x
+63660y+600xy+300
2
x
y+51504
2
y
-62378
2
x
2
y
-14154
3
y
+1500x
3
y
-54010
2
x
3
y
-10130
4
y
-12050
2
x
4
y
-17400
5
y
+1250
8
y
+2140z-1200xz+2064
2
x
z-608yz+520
2
x
yz-240
2
y
z-14000x
2
y
z-5800x
3
y
z-384
2
z
+6500x
2
z
-1200
2
x
2
z
+480y
2
z
+1500xy
2
z
+29260
2
y
2
z
+35004
3
y
2
z
+9520
4
y
2
z
-33800
3
z
+1660
2
x
3
z
-26020y
3
z
-100
2
x
y
3
z
-4800
2
y
3
z
-1600x
2
y
3
z
+500x
3
y
3
z
},{{0,0,1},{-1,0,0},{1,1,z},{-x+5
2
y
,-x,1-xz+5
2
y
z},{30+8
2
y
-25
2
x
2
y
-5
3
y
-4z+20
2
y
2
z
-10
3
z
,-4z-10
3
z
,25x
2
y
-4
2
z
-10
4
z
},{13770+200x-1500
2
x
-625
3
x
+4800y-1128
2
y
-8750
2
x
2
y
+1250
4
x
2
y
-2135
3
y
-3750
2
x
3
y
-800
4
y
+64z+200
2
x
z-80yz-2000x
2
y
z+500x
2
z
+4180
2
y
2
z
-1000
2
x
2
y
2
z
+3200
3
y
2
z
-4590
3
z
+500
2
x
3
z
-1600y
3
z
,3750+400x-625
3
x
-625
3
y
+564z+200
2
x
z-80yz+1000x
2
z
-4590
3
z
+500
2
x
3
z
-1600y
3
z
,6050+1500x+625
2
x
+1875y-7500
2
y
-1250
3
x
2
y
+1250
5
y
+200xz-625
3
x
z+3125
2
x
2
y
z+64
2
z
+200
2
x
2
z
-80y
2
z
+500x
3
z
-5000
2
y
3
z
-4590
4
z
+500
2
x
4
z
-1600y
4
z
}}}
Out[]=
Check that the bases agree up to constant factors (they are not identical in this instance):
Together[gb1/gb2]
In[]:=
1,1,1,1,1,
1
2

Out[]=
Check the conversion matrix identity:
Expand[cmat.polys-gb2]
In[]:=

Properties and Relations

Compute the extended GCD:
Check the Bezout relation for the extended GCD:
Again check the Bezout relation:
Now check that these two computations agree up to a constant factor:

Possible Issues