-- -*- M2-comint -*- {* hash: 1728887545 *}
i1 : R = ZZ/1277[x,y];
i2 : I = ideal(x^3 - 2*x*y, x^2*y - 2*y^2 + x);
o2 : Ideal of R
i3 : g = gb I
o3 = GroebnerBasis[status: done; S-pairs encountered up to degree 5]
o3 : GroebnerBasis
i4 : gens g
o4 = | y2+638x xy x2 |
1 3
o4 : Matrix R <--- R
i5 : R = ZZ/1277[x,y,z,w];
i6 : I = ideal(x*y-z^2,y^2-w^2);
o6 : Ideal of R
i7 : g2 = gb(I,DegreeLimit => 2)
o7 = GroebnerBasis[status: DegreeLimit; all S-pairs handled up to degree 2]
o7 : GroebnerBasis
i8 : gens g2
o8 = | y2-w2 xy-z2 |
1 2
o8 : Matrix R <--- R
i9 : g3 = gb(I,DegreeLimit => 3);
i10 : gens g3
o10 = | y2-w2 xy-z2 yz2-xw2 |
1 3
o10 : Matrix R <--- R
i11 : g2
o11 = GroebnerBasis[status: DegreeLimit; all S-pairs handled up to degree 3]
o11 : GroebnerBasis
i12 : g2 === g3
o12 = true
i13 : I = ideal(x*y-z^2,y^2-w^2)
2 2 2
o13 = ideal (x*y - z , y - w )
o13 : Ideal of R
i14 : gb(I,PairLimit => 2)
o14 = GroebnerBasis[status: PairLimit; all S-pairs handled up to degree 1]
o14 : GroebnerBasis
i15 : gb(I,PairLimit => 3)
o15 = GroebnerBasis[status: PairLimit; all S-pairs handled up to degree 2]
o15 : GroebnerBasis
i16 : I = ideal(x*y-z^2,y^2-w^2)
2 2 2
o16 = ideal (x*y - z , y - w )
o16 : Ideal of R
i17 : gb(I,BasisElementLimit => 2)
o17 = GroebnerBasis[status: BasisElementLimit; all S-pairs handled up to degree 1]
o17 : GroebnerBasis
i18 : gb(I,BasisElementLimit => 3)
o18 = GroebnerBasis[status: BasisElementLimit; all S-pairs handled up to degree 2]
o18 : GroebnerBasis
i19 : R = ZZ/1277[t,F,G,MonomialOrder => Eliminate 1];
i20 : I = ideal(F - (t^3 + t^2 + 1), G - (t^4 - t))
3 2 4
o20 = ideal (- t - t + F - 1, - t + t + G)
o20 : Ideal of R
i21 : transpose gens gb (I, SubringLimit => 1)
o21 = {-4} | F4-7F3-2F2G-4FG2-G3+18F2+3FG+6G2-21F-G+9 |
{-3} | tG2-tF+6tG+5t-F3+3F2+3FG+3G2+G-2 |
{-3} | tFG+tF-4tG-3t+F2-FG-G2-4F-G+3 |
{-3} | tF2-4tF+tG+5t-F2-FG+3F+3G-2 |
{-2} | t2+tF-2t-F-G+1 |
5 1
o21 : Matrix R <--- R
i22 : gbTrace = 3
o22 = 3
i23 : I = ideal(x*y-z^2,y^2-w^2)
2 2 2
o23 = ideal (x*y - z , y - w )
ZZ
o23 : Ideal of ----[x, y, z, w]
1277
i24 : gb I
-- registering gb 6 at 0x933e260
-- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)o
-- number of (nonminimal) gb elements = 4
-- number of monomials = 8
-- ncalls = 0
-- nloop = 0
-- nsaved = 0
--
o24 = GroebnerBasis[status: done; S-pairs encountered up to degree 4]
o24 : GroebnerBasis
i25 : gbTrace = 0
o25 = 0
i26 : R = ZZ/1277[x..z];
i27 : I = ideal(x*y+y*z, y^2, x^2);
o27 : Ideal of R
i28 : g = gb(I, StopBeforeComputation => true)
o28 = GroebnerBasis[status: not started; all S-pairs handled up to degree -1]
o28 : GroebnerBasis
i29 : gens g
o29 = 0
1
o29 : Matrix R <--- 0
i30 : R = ZZ/1277[a..e];
i31 : T = (degreesRing R)_0
o31 = T
o31 : ZZ[T]
i32 : f = random(R^1,R^{-3,-3,-5,-6});
1 4
o32 : Matrix R <--- R
i33 : time betti gb f
-- used 0.290956 seconds
0 1
o33 = total: 1 53
0: 1 .
1: . .
2: . 2
3: . 1
4: . 2
5: . 3
6: . 5
7: . 5
8: . 8
9: . 9
10: . 8
11: . 6
12: . 3
13: . 1
o33 : BettiTally
i34 : remove(f.cache,{false,0})
i35 : (cokernel f).cache.poincare = (1-T^3)*(1-T^3)*(1-T^5)*(1-T^6)
3 5 8 9 12 14 17
o35 = 1 - 2T - T + 2T + 2T - T - 2T + T
o35 : ZZ[T]
i36 : time betti gb f
-- used 0.004 seconds
0 1
o36 = total: 1 53
0: 1 .
1: . .
2: . 2
3: . 1
4: . 2
5: . 3
6: . 5
7: . 5
8: . 8
9: . 9
10: . 8
11: . 6
12: . 3
13: . 1
o36 : BettiTally
i37 :
