Let's consider the problem of finding the polynomial relation of the three polynomials $x = s^3+st+1$, $y = t^3+3t^2+t$, and $z = st^3$. The first method we use is to compute a Groebner basis using an elimination order which eliminates the variables s,t, and then select those Groebner basis elements that do not involve the variables s and t.
i1 : R = QQ[s,t,x,y,z, MonomialOrder=>Eliminate 2];
|
i2 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
3 3 2 3
o2 = ideal (- s - s*t + x - 1, - t - 3t - t + y, - s*t + z)
o2 : Ideal of R
|
i3 : time leadTerm gens gb I
-- used 0.215024 seconds
o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
------------------------------------------------------------------------
563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
------------------------------------------------------------------------
267076255345488270sy3z4 5256861933965245618410txyz6
------------------------------------------------------------------------
259806064190048723110954950tx2z6 281231296878799148310ty3z5
------------------------------------------------------------------------
178193459663956600213275sx2yz5 2019525876191508135750450sy2z7
------------------------------------------------------------------------
3470026149887950992631901569317758202349494375sx2z8 4004st 12012t2z
------------------------------------------------------------------------
3003s2z 9009t2x 14s2y3 t3 4004s3 |
1 25
o3 : Matrix R <-- R
|
i4 : G = selectInSubring(1,gens gb I)
o4 = | x3y9-3x2y9-6x2y8z-3x2y6z3+3xy9-x2y7z+12xy8z+7xy7z2-324x2y5z3+6xy6z3-
------------------------------------------------------------------------
y7z3-15xy5z4+3xy3z6-y9+2xy7z-6y8z-7y7z2-1233x2y4z3+648xy5z3-3y6z3+
------------------------------------------------------------------------
440xy4z4+15y5z4-70y4z5-405xy2z6-3y3z6+21y2z7-z9-y7z-945x2y3z3+2466xy4z3-
------------------------------------------------------------------------
324y5z3+1749xy3z4-440y4z4-252y3z5+3393xyz6+405y2z6-385yz7-252x2y2z3+
------------------------------------------------------------------------
1890xy3z3-1233y4z3+930xy2z4-1749y3z4-133y2z5-5778xz6-3393yz6+843z7-
------------------------------------------------------------------------
27x2yz3+504xy2z3-945y3z3+147xyz4-930y2z4-21yz5+5778z6-x2z3+54xyz3-
------------------------------------------------------------------------
252y2z3+7xz4-147yz4-z5+2xz3-27yz3-7z4-z3 |
1 1
o4 : Matrix R <-- R
|
i5 : ans1 = G_(0,0)
3 9 2 9 2 8 2 6 3 9 2 7 8 7 2
o5 = x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z + 7x*y z -
------------------------------------------------------------------------
2 5 3 6 3 7 3 5 4 3 6 9 7 8
324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z - 6y z -
------------------------------------------------------------------------
7 2 2 4 3 5 3 6 3 4 4 5 4 4 5
7y z - 1233x y z + 648x*y z - 3y z + 440x*y z + 15y z - 70y z -
------------------------------------------------------------------------
2 6 3 6 2 7 9 7 2 3 3 4 3 5 3
405x*y z - 3y z + 21y z - z - y z - 945x y z + 2466x*y z - 324y z
------------------------------------------------------------------------
3 4 4 4 3 5 6 2 6 7
+ 1749x*y z - 440y z - 252y z + 3393x*y*z + 405y z - 385y*z -
------------------------------------------------------------------------
2 2 3 3 3 4 3 2 4 3 4 2 5
252x y z + 1890x*y z - 1233y z + 930x*y z - 1749y z - 133y z -
------------------------------------------------------------------------
6 6 7 2 3 2 3 3 3 4
5778x*z - 3393y*z + 843z - 27x y*z + 504x*y z - 945y z + 147x*y*z
------------------------------------------------------------------------
2 4 5 6 2 3 3 2 3 4
- 930y z - 21y*z + 5778z - x z + 54x*y*z - 252y z + 7x*z -
------------------------------------------------------------------------
4 5 3 3 4 3
147y*z - z + 2x*z - 27y*z - 7z - z
o5 : R
|
This method (with some optimizations incorporated) is provided by the
eliminate function.
i6 : R = QQ[x,y,z,s,t];
|
i7 : I = ideal(x-s^3-s*t-1, y-t^3-3*t^2-t, z-s*t^3)
3 3 2 3
o7 = ideal (- s - s*t + x - 1, - t - 3t + y - t, - s*t + z)
o7 : Ideal of R
|
i8 : time G = eliminate(I,{s,t})
-- used 0.212816 seconds
3 9 2 9 2 8 2 6 3 9 2 7 8
o8 = ideal(x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z +
------------------------------------------------------------------------
7 2 2 5 3 6 3 7 3 5 4 3 6 9 7
7x*y z - 324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z
------------------------------------------------------------------------
8 7 2 2 4 3 5 3 6 3 4 4 5 4
- 6y z - 7y z - 1233x y z + 648x*y z - 3y z + 440x*y z + 15y z -
------------------------------------------------------------------------
4 5 2 6 3 6 2 7 9 7 2 3 3 4 3
70y z - 405x*y z - 3y z + 21y z - z - y z - 945x y z + 2466x*y z
------------------------------------------------------------------------
5 3 3 4 4 4 3 5 6 2 6
- 324y z + 1749x*y z - 440y z - 252y z + 3393x*y*z + 405y z -
------------------------------------------------------------------------
7 2 2 3 3 3 4 3 2 4 3 4
385y*z - 252x y z + 1890x*y z - 1233y z + 930x*y z - 1749y z -
------------------------------------------------------------------------
2 5 6 6 7 2 3 2 3 3 3
133y z - 5778x*z - 3393y*z + 843z - 27x y*z + 504x*y z - 945y z +
------------------------------------------------------------------------
4 2 4 5 6 2 3 3 2 3
147x*y*z - 930y z - 21y*z + 5778z - x z + 54x*y*z - 252y z +
------------------------------------------------------------------------
4 4 5 3 3 4 3
7x*z - 147y*z - z + 2x*z - 27y*z - 7z - z )
o8 : Ideal of R
|
i9 : ans2 = G_0
3 9 2 9 2 8 2 6 3 9 2 7 8 7 2
o9 = x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z + 7x*y z -
------------------------------------------------------------------------
2 5 3 6 3 7 3 5 4 3 6 9 7 8
324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z - 6y z -
------------------------------------------------------------------------
7 2 2 4 3 5 3 6 3 4 4 5 4 4 5
7y z - 1233x y z + 648x*y z - 3y z + 440x*y z + 15y z - 70y z -
------------------------------------------------------------------------
2 6 3 6 2 7 9 7 2 3 3 4 3 5 3
405x*y z - 3y z + 21y z - z - y z - 945x y z + 2466x*y z - 324y z
------------------------------------------------------------------------
3 4 4 4 3 5 6 2 6 7
+ 1749x*y z - 440y z - 252y z + 3393x*y*z + 405y z - 385y*z -
------------------------------------------------------------------------
2 2 3 3 3 4 3 2 4 3 4 2 5
252x y z + 1890x*y z - 1233y z + 930x*y z - 1749y z - 133y z -
------------------------------------------------------------------------
6 6 7 2 3 2 3 3 3 4
5778x*z - 3393y*z + 843z - 27x y*z + 504x*y z - 945y z + 147x*y*z
------------------------------------------------------------------------
2 4 5 6 2 3 3 2 3 4
- 930y z - 21y*z + 5778z - x z + 54x*y*z - 252y z + 7x*z -
------------------------------------------------------------------------
4 5 3 3 4 3
147y*z - z + 2x*z - 27y*z - 7z - z
o9 : R
|
Sometimes giving the variables different degrees will speed up the computations. Here, we set the degrees of x, y, and z to be the total degrees.
i10 : R1 = QQ[x,y,z,s,t, Degrees=>{3,3,4,1,1}];
|
i11 : I1 = substitute(I,R1);
o11 : Ideal of R1
|
i12 : time G = eliminate(I1,{s,t})
-- used 0.0589559 seconds
3 9 2 6 3 3 6 9 2 8 5 4 2 7
o12 = ideal(x y - 3x y z + 3x*y z - z - 6x y z - 15x*y z + 21y z -
-----------------------------------------------------------------------
2 9 2 5 3 6 3 7 3 2 6 3 6 7 2
3x y - 324x y z + 6x*y z - y z - 405x*y z - 3y z + 7x*y z -
-----------------------------------------------------------------------
4 5 2 7 8 4 4 5 4 7 9
70y z - x y z + 12x*y z + 440x*y z + 15y z - 385y*z + 3x*y -
-----------------------------------------------------------------------
2 4 3 5 3 6 3 6 2 6 7 2 3 5
1233x y z + 648x*y z - 3y z + 3393x*y*z + 405y z - 7y z - 252y z
-----------------------------------------------------------------------
7 8 3 4 4 4 7 9 2 3 3
+ 2x*y z - 6y z + 1749x*y z - 440y z + 843z - y - 945x y z +
-----------------------------------------------------------------------
4 3 5 3 6 6 2 5 7 2 4
2466x*y z - 324y z - 5778x*z - 3393y*z - 133y z - y z + 930x*y z
-----------------------------------------------------------------------
3 4 2 2 3 3 3 4 3 6 5
- 1749y z - 252x y z + 1890x*y z - 1233y z + 5778z - 21y*z +
-----------------------------------------------------------------------
4 2 4 2 3 2 3 3 3 5 4
147x*y*z - 930y z - 27x y*z + 504x*y z - 945y z - z + 7x*z -
-----------------------------------------------------------------------
4 2 3 3 2 3 4 3 3 3
147y*z - x z + 54x*y*z - 252y z - 7z + 2x*z - 27y*z - z )
o12 : Ideal of R1
|
i13 : ans3 = G_0
3 9 2 6 3 3 6 9 2 8 5 4 2 7 2 9
o13 = x y - 3x y z + 3x*y z - z - 6x y z - 15x*y z + 21y z - 3x y -
-----------------------------------------------------------------------
2 5 3 6 3 7 3 2 6 3 6 7 2 4 5
324x y z + 6x*y z - y z - 405x*y z - 3y z + 7x*y z - 70y z -
-----------------------------------------------------------------------
2 7 8 4 4 5 4 7 9 2 4 3
x y z + 12x*y z + 440x*y z + 15y z - 385y*z + 3x*y - 1233x y z +
-----------------------------------------------------------------------
5 3 6 3 6 2 6 7 2 3 5 7
648x*y z - 3y z + 3393x*y*z + 405y z - 7y z - 252y z + 2x*y z -
-----------------------------------------------------------------------
8 3 4 4 4 7 9 2 3 3 4 3
6y z + 1749x*y z - 440y z + 843z - y - 945x y z + 2466x*y z -
-----------------------------------------------------------------------
5 3 6 6 2 5 7 2 4 3 4
324y z - 5778x*z - 3393y*z - 133y z - y z + 930x*y z - 1749y z -
-----------------------------------------------------------------------
2 2 3 3 3 4 3 6 5 4
252x y z + 1890x*y z - 1233y z + 5778z - 21y*z + 147x*y*z -
-----------------------------------------------------------------------
2 4 2 3 2 3 3 3 5 4 4 2 3
930y z - 27x y*z + 504x*y z - 945y z - z + 7x*z - 147y*z - x z
-----------------------------------------------------------------------
3 2 3 4 3 3 3
+ 54x*y*z - 252y z - 7z + 2x*z - 27y*z - z
o13 : R1
|
Another approach is to create the ring map $F : k[x,y,z] \rightarrow{} k[s,t]$, and find its kernel.
i14 : A = QQ[s,t];
|
i15 : B = QQ[x,y,z];
|
i16 : F = map(A,B,{s^3+s*t+1, t^3+3*t^2+t, s*t^3})
3 3 2 3
o16 = map (A, B, {s + s*t + 1, t + 3t + t, s*t })
o16 : RingMap A <-- B
|
i17 : time G = kernel F
-- used 0.20996 seconds
3 9 2 9 2 8 2 6 3 9 2 7 8
o17 = ideal(x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z +
-----------------------------------------------------------------------
7 2 2 5 3 6 3 7 3 5 4 3 6 9 7
7x*y z - 324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z
-----------------------------------------------------------------------
8 7 2 2 4 3 5 3 6 3 4 4 5 4
- 6y z - 7y z - 1233x y z + 648x*y z - 3y z + 440x*y z + 15y z -
-----------------------------------------------------------------------
4 5 2 6 3 6 2 7 9 7 2 3 3 4 3
70y z - 405x*y z - 3y z + 21y z - z - y z - 945x y z + 2466x*y z
-----------------------------------------------------------------------
5 3 3 4 4 4 3 5 6 2 6
- 324y z + 1749x*y z - 440y z - 252y z + 3393x*y*z + 405y z -
-----------------------------------------------------------------------
7 2 2 3 3 3 4 3 2 4 3 4
385y*z - 252x y z + 1890x*y z - 1233y z + 930x*y z - 1749y z -
-----------------------------------------------------------------------
2 5 6 6 7 2 3 2 3 3 3
133y z - 5778x*z - 3393y*z + 843z - 27x y*z + 504x*y z - 945y z
-----------------------------------------------------------------------
4 2 4 5 6 2 3 3 2 3
+ 147x*y*z - 930y z - 21y*z + 5778z - x z + 54x*y*z - 252y z +
-----------------------------------------------------------------------
4 4 5 3 3 4 3
7x*z - 147y*z - z + 2x*z - 27y*z - 7z - z )
o17 : Ideal of B
|
i18 : ans4 = G_0
3 9 2 9 2 8 2 6 3 9 2 7 8 7 2
o18 = x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z + 7x*y z -
-----------------------------------------------------------------------
2 5 3 6 3 7 3 5 4 3 6 9 7 8
324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z - 6y z -
-----------------------------------------------------------------------
7 2 2 4 3 5 3 6 3 4 4 5 4 4 5
7y z - 1233x y z + 648x*y z - 3y z + 440x*y z + 15y z - 70y z -
-----------------------------------------------------------------------
2 6 3 6 2 7 9 7 2 3 3 4 3
405x*y z - 3y z + 21y z - z - y z - 945x y z + 2466x*y z -
-----------------------------------------------------------------------
5 3 3 4 4 4 3 5 6 2 6
324y z + 1749x*y z - 440y z - 252y z + 3393x*y*z + 405y z -
-----------------------------------------------------------------------
7 2 2 3 3 3 4 3 2 4 3 4
385y*z - 252x y z + 1890x*y z - 1233y z + 930x*y z - 1749y z -
-----------------------------------------------------------------------
2 5 6 6 7 2 3 2 3 3 3
133y z - 5778x*z - 3393y*z + 843z - 27x y*z + 504x*y z - 945y z
-----------------------------------------------------------------------
4 2 4 5 6 2 3 3 2 3
+ 147x*y*z - 930y z - 21y*z + 5778z - x z + 54x*y*z - 252y z +
-----------------------------------------------------------------------
4 4 5 3 3 4 3
7x*z - 147y*z - z + 2x*z - 27y*z - 7z - z
o18 : B
|
This appears to be much faster than the first two methods.
Finally, we may use resultants to find elements of the ideal I which do not involve the variables s and t.
i19 : use ring I
o19 = R
o19 : PolynomialRing
|
i20 : time f1 = resultant(I_0,I_2,s)
-- used 0.00116117 seconds
9 9 7 3
o20 = x*t - t - z*t - z
o20 : R
|
i21 : time f2 = resultant(I_1,f1,t)
-- used 0.0816464 seconds
3 9 2 9 2 8 2 6 3 9 2 7 8 7 2
o21 = - x y + 3x y + 6x y z + 3x y z - 3x*y + x y z - 12x*y z - 7x*y z +
-----------------------------------------------------------------------
2 5 3 6 3 7 3 5 4 3 6 9 7 8
324x y z - 6x*y z + y z + 15x*y z - 3x*y z + y - 2x*y z + 6y z +
-----------------------------------------------------------------------
7 2 2 4 3 5 3 6 3 4 4 5 4 4 5
7y z + 1233x y z - 648x*y z + 3y z - 440x*y z - 15y z + 70y z +
-----------------------------------------------------------------------
2 6 3 6 2 7 9 7 2 3 3 4 3
405x*y z + 3y z - 21y z + z + y z + 945x y z - 2466x*y z +
-----------------------------------------------------------------------
5 3 3 4 4 4 3 5 6 2 6
324y z - 1749x*y z + 440y z + 252y z - 3393x*y*z - 405y z +
-----------------------------------------------------------------------
7 2 2 3 3 3 4 3 2 4 3 4
385y*z + 252x y z - 1890x*y z + 1233y z - 930x*y z + 1749y z +
-----------------------------------------------------------------------
2 5 6 6 7 2 3 2 3 3 3
133y z + 5778x*z + 3393y*z - 843z + 27x y*z - 504x*y z + 945y z
-----------------------------------------------------------------------
4 2 4 5 6 2 3 3 2 3
- 147x*y*z + 930y z + 21y*z - 5778z + x z - 54x*y*z + 252y z -
-----------------------------------------------------------------------
4 4 5 3 3 4 3
7x*z + 147y*z + z - 2x*z + 27y*z + 7z + z
o21 : R
|
i22 : ans5 = -f2
3 9 2 9 2 8 2 6 3 9 2 7 8 7 2
o22 = x y - 3x y - 6x y z - 3x y z + 3x*y - x y z + 12x*y z + 7x*y z -
-----------------------------------------------------------------------
2 5 3 6 3 7 3 5 4 3 6 9 7 8
324x y z + 6x*y z - y z - 15x*y z + 3x*y z - y + 2x*y z - 6y z -
-----------------------------------------------------------------------
7 2 2 4 3 5 3 6 3 4 4 5 4 4 5
7y z - 1233x y z + 648x*y z - 3y z + 440x*y z + 15y z - 70y z -
-----------------------------------------------------------------------
2 6 3 6 2 7 9 7 2 3 3 4 3
405x*y z - 3y z + 21y z - z - y z - 945x y z + 2466x*y z -
-----------------------------------------------------------------------
5 3 3 4 4 4 3 5 6 2 6
324y z + 1749x*y z - 440y z - 252y z + 3393x*y*z + 405y z -
-----------------------------------------------------------------------
7 2 2 3 3 3 4 3 2 4 3 4
385y*z - 252x y z + 1890x*y z - 1233y z + 930x*y z - 1749y z -
-----------------------------------------------------------------------
2 5 6 6 7 2 3 2 3 3 3
133y z - 5778x*z - 3393y*z + 843z - 27x y*z + 504x*y z - 945y z
-----------------------------------------------------------------------
4 2 4 5 6 2 3 3 2 3
+ 147x*y*z - 930y z - 21y*z + 5778z - x z + 54x*y*z - 252y z +
-----------------------------------------------------------------------
4 4 5 3 3 4 3
7x*z - 147y*z - z + 2x*z - 27y*z - 7z - z
o22 : R
|
This is the fastest method in this case.
These answers should all be the same (with the possible exception of the last), but are they? They live in different rings, so we cannot compare them directly. Instead, let's move them to the ring B, and then remove duplicates.
i23 : L = {ans1,ans2,ans3,ans4,ans5};
|
i24 : L = apply(L, f -> substitute(f,B));
|
i25 : length unique L
o25 = 1
|
They are all the same!