-- -*- M2-comint -*- {* hash: -2037591551 *}
i1 : KK = ZZ/32003
o1 = KK
o1 : QuotientRing
i2 : R = KK[x,y,z,w]
o2 = R
o2 : PolynomialRing
i3 : I = ideal(x^2*y,x*y^2+x^3)
2 3 2
o3 = ideal (x y, x + x*y )
o3 : Ideal of R
i4 : J = gens gb I
o4 = | x2y x3+xy2 xy3 |
1 3
o4 : Matrix R <--- R
i5 : R = KK[x,y,z,w,MonomialOrder=>Lex]
o5 = R
o5 : PolynomialRing
i6 : I = substitute(I,R)
2 3 2
o6 = ideal (x y, x + x*y )
o6 : Ideal of R
i7 : gens gb I
o7 = | xy3 x2y x3+xy2 |
1 3
o7 : Matrix R <--- R
i8 : R = KK[x,y,z]
o8 = R
o8 : PolynomialRing
i9 : F = random(R^1, R^{-2,-3})
o9 = | 4784x2+7336xy+5204y2-13630xz-5803yz+178z2
------------------------------------------------------------------------
4812x3-5078x2y+1744xy2-8124y3+711x2z+12462xyz-1117y2z+7264xz2-3703yz2+
------------------------------------------------------------------------
13609z3 |
1 2
o9 : Matrix R <--- R
i10 : GB = gens gb F
o10 = | x2+15896xy-12281y2-12887xz+6481yz+281z2
-----------------------------------------------------------------------
xy2+2071y3-14824xyz-3889y2z-11159xz2+11464yz2-12986z3
-----------------------------------------------------------------------
y4+1199y3z-14666xyz2+15864y2z2-12070xz3-8899yz3-13623z4 |
1 3
o10 : Matrix R <--- R
i11 : LT = leadTerm GB
o11 = | x2 xy2 y4 |
1 3
o11 : Matrix R <--- R
i12 : betti LT
0 1
o12 = total: 1 3
0: 1 .
1: . 1
2: . 1
3: . 1
o12 : BettiTally
i13 : R = KK[x,y,z, MonomialOrder => Lex]
o13 = R
o13 : PolynomialRing
i14 : F = random(R^1, R^{-2,-3})
o14 = | -9607x2-2779xy-6990xz-13967y2-8493yz+4601z2
-----------------------------------------------------------------------
-3927x3+8362x2y+2217x2z+1456xy2+5336xyz-2384xz2-15285y3-15520y2z+
-----------------------------------------------------------------------
7367yz2+14266z3 |
1 2
o14 : Matrix R <--- R
i15 : GB = gens gb F
o15 = | y6-10565y5z-2181y4z2+15036y3z3-8915y2z4-125yz5-13748z6
-----------------------------------------------------------------------
xz4-5444y5+10927y4z-635y3z2+5108y2z3-922yz4-13864z5
-----------------------------------------------------------------------
xyz2-2574xz3+3653y4-245y3z-12393y2z2+14708yz3+13496z4
-----------------------------------------------------------------------
xy2+15472xyz-1817xz2-9110y3-2993y2z-13510yz2+9710z3
-----------------------------------------------------------------------
x2+11473xy-13544xz-4006y2+13459yz+13411z2 |
1 5
o15 : Matrix R <--- R
i16 : LT = leadTerm GB
o16 = | y6 xz4 xyz2 xy2 x2 |
1 5
o16 : Matrix R <--- R
i17 : betti LT
0 1
o17 = total: 1 5
0: 1 .
1: . 1
2: . 1
3: . 1
4: . 1
5: . 1
o17 : BettiTally
i18 : R = KK[a..i]
o18 = R
o18 : PolynomialRing
i19 : M = genericMatrix(R,a,3,3)
o19 = | a d g |
| b e h |
| c f i |
3 3
o19 : Matrix R <--- R
i20 : N = M^3
o20 = | a3+2abd+bde+2acg+bfg+cdh+cgi
| a2b+b2d+abe+be2+bcg+ach+ceh+bfh+chi
| a2c+bcd+abf+bef+c2g+cfh+aci+bfi+ci2
-----------------------------------------------------------------------
a2d+bd2+ade+de2+cdg+afg+efg+dfh+fgi a2g+bdg+cg2+adh+deh+fgh+agi+dhi+gi2
abd+2bde+e3+bfg+cdh+2efh+fhi abg+beg+bdh+e2h+cgh+fh2+bgi+ehi+hi2
acd+cde+bdf+e2f+cfg+f2h+cdi+efi+fi2 acg+bfg+cdh+efh+2cgi+2fhi+i3
-----------------------------------------------------------------------
|
|
|
3 3
o20 : Matrix R <--- R
i21 : I = flatten N
o21 = | a3+2abd+bde+2acg+bfg+cdh+cgi a2b+b2d+abe+be2+bcg+ach+ceh+bfh+chi
-----------------------------------------------------------------------
a2c+bcd+abf+bef+c2g+cfh+aci+bfi+ci2 a2d+bd2+ade+de2+cdg+afg+efg+dfh+fgi
-----------------------------------------------------------------------
abd+2bde+e3+bfg+cdh+2efh+fhi acd+cde+bdf+e2f+cfg+f2h+cdi+efi+fi2
-----------------------------------------------------------------------
a2g+bdg+cg2+adh+deh+fgh+agi+dhi+gi2 abg+beg+bdh+e2h+cgh+fh2+bgi+ehi+hi2
-----------------------------------------------------------------------
acg+bfg+cdh+efh+2cgi+2fhi+i3 |
1 9
o21 : Matrix R <--- R
i22 : Tr = trace M
o22 = a + e + i
o22 : R
i23 : Tr //I -- the quotient, which is 0
o23 = 0
9 1
o23 : Matrix R <--- R
i24 : Tr % I -- the remainder, which is Tr again
o24 = a + e + i
o24 : R
i25 : Tr^2 % I
2 2 2
o25 = a + 2a*e + e + 2a*i + 2e*i + i
o25 : R
i26 : Tr^3 % I
2 2 3 2
o26 = 3a e + 3b*d*e + 3a*e + 3e + 3b*f*g + 3c*d*h + 6e*f*h + 3a i + 6a*e*i
-----------------------------------------------------------------------
2 2 2 3
+ 3e i + 3c*g*i + 6f*h*i + 3a*i + 3e*i + 3i
o26 : R
i27 : Tr^4 % I
2 2 2 3 4
o27 = 6a e + 6b*d*e + 6a*e + 6e + 6b*e*f*g + 6c*d*e*h - 6b*d*f*h +
-----------------------------------------------------------------------
2 2 2 2 2
6a*e*f*h + 12e f*h - 6c*f*g*h - 6f h + 12a e*i + 12b*d*e*i + 12a*e i +
-----------------------------------------------------------------------
3 2 2
12e i + 12c*e*g*i + 6b*f*g*i + 6c*d*h*i + 6a*f*h*i + 36e*f*h*i + 6a i
-----------------------------------------------------------------------
2 2 2 2 2 3 3 4
+ 12a*e*i + 6e i + 6c*g*i + 12f*h*i + 6a*i + 12e*i + 6i
o27 : R
i28 : Tr^5 % I
2 2 2 3 4 2 2
o28 = 30a e i + 30b*d*e i + 30a*e i + 30e i + 30c*e g*i + 30a f*h*i +
-----------------------------------------------------------------------
2 2 2
30b*d*f*h*i + 60a*e*f*h*i + 90e f*h*i + 30c*f*g*h*i + 30f h i +
-----------------------------------------------------------------------
2 2 2 2 2 3 2 2 2
30a e*i + 30b*d*e*i + 30a*e i + 30e i + 30c*e*g*i + 60a*f*h*i +
-----------------------------------------------------------------------
2 2 3 3 3 2 3 3
120e*f*h*i + 30a i + 30b*d*i + 30a*e*i + 30e i + 30c*g*i +
-----------------------------------------------------------------------
3 4 4 5
90f*h*i + 30a*i + 30e*i + 30i
o28 : R
i29 : Tr^6 % I
2 2 2 2 2 3 2 4 2 2 2 2 2
o29 = 90a e i + 90b*d*e i + 90a*e i + 90e i + 90c*e g*i + 90a f*h*i +
-----------------------------------------------------------------------
2 2 2 2 2 2 2 2
90b*d*f*h*i + 180a*e*f*h*i + 270e f*h*i + 90c*f*g*h*i + 90f h i +
-----------------------------------------------------------------------
2 3 3 2 3 3 3 3 3
90a e*i + 90b*d*e*i + 90a*e i + 90e i + 90c*e*g*i + 180a*f*h*i +
-----------------------------------------------------------------------
3 2 4 4 4 2 4 4
360e*f*h*i + 90a i + 90b*d*i + 90a*e*i + 90e i + 90c*g*i +
-----------------------------------------------------------------------
4 5 5 6
270f*h*i + 90a*i + 90e*i + 90i
o29 : R
i30 : Tr^7 % I
o30 = 0
o30 : R
i31 : Tr^6 // I
o31 = {3} | a3+6a2e+3bde+15ae2+22e3+6efh+6a2i+30aei+60e2i+6fhi+15ai2+60ei2+22
{3} | 0
{3} | 0
{3} | -27abe-45be2+9ceh+30bfh-72abi-144bei+75chi
{3} | -2a3+15a2e+21bde+6ae2+e3+33bfg+9cdh-36afh+51efh+60a2i+72bdi+30aei
{3} | 18abg+45beg+3a2h+9bdh-21aeh+3e2h+9cgh+36fh2+114bgi-135ahi-39ehi+3
{3} | 18ace+6abf-36bef-18cfh+66aci+36cei-57bfi+132ci2
{3} | -3a2f-39bdf+75aef-12e2f+9cfg-36f2h-66cdi+135afi+51efi+69fi2
{3} | -2a3-18abd-30a2e-60bde-30ae2-44e3-18ceg-33bfg-9cdh+18afh-93efh-75
-----------------------------------------------------------------------
i3 |
|
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+6e2i+66cgi+147fhi-30ai2-75ei2-26i3 |
6hi2 |
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a2i-90bdi-60aei-75e2i-66cgi-171fhi-84ai2-84ei2-89i3 |
9 1
o31 : Matrix R <--- R
i32 : Tr^6 == I * (Tr^6 // I) + (Tr^6 % I)
o32 = true
i33 : R = KK[t,y,z,MonomialOrder=>Lex]
o33 = R
o33 : PolynomialRing
i34 : I = ideal(y-(t^2+t+1), z-(t^3+1))
2 3
o34 = ideal (- t - t + y - 1, - t + z - 1)
o34 : Ideal of R
i35 : GB = gens gb I
o35 = | y3-3y2-3yz+6y-z2+4z-4 tz-2t-y2+3y+2z-4 ty-y-z+2 t2+t-y+1 |
1 4
o35 : Matrix R <--- R
i36 : F = GB_(0,0)
3 2 2
o36 = y - 3y - 3y*z + 6y - z + 4z - 4
o36 : R
i37 : substitute(F, {y =>t^2+t+1, z=>t^3+1})
o37 = 0
o37 : R
i38 : R = KK[y,z,t]
o38 = R
o38 : PolynomialRing
i39 : I = substitute(I,R)
2 3
o39 = ideal (- t + y - t - 1, - t + z - 1)
o39 : Ideal of R
i40 : eliminate(I,t)
3 2 2
o40 = ideal(y - 3y - 3y*z - z + 6y + 4z - 4)
o40 : Ideal of R
i41 : A = KK[t]
o41 = A
o41 : PolynomialRing
i42 : B = KK[y,z]
o42 = B
o42 : PolynomialRing
i43 : G = map(A,B,{t^2+t+1, t^3+1})
2 3
o43 = map(A,B,{t + t + 1, t + 1})
o43 : RingMap A <--- B
i44 : kernel G
3 2 2
o44 = ideal(y - 3y - 3y*z - z + 6y + 4z - 4)
o44 : Ideal of B
i45 : R = KK[t,x,y,z]
o45 = R
o45 : PolynomialRing
i46 : I = ideal(x^3,y^3,z^3)
3 3 3
o46 = ideal (x , y , z )
o46 : Ideal of R
i47 : F = x+y+z
o47 = x + y + z
o47 : R
i48 : L = t*I + (1-t)*ideal(F)
3 3 3
o48 = ideal (t*x , t*y , t*z , - t*x - t*y - t*z + x + y + z)
o48 : Ideal of R
i49 : L1 = eliminate(L,t)
3 3 4 3 4 3 3 4 3 3 3
o49 = ideal (x*z + y*z + z , x*y + y + y z, x y + y - x z + 2y z - 2y*z
-----------------------------------------------------------------------
4 4 4 3 3 3 4 3 2 3 2 2 3 4
- z , x - y + 2x z - 2y z + 2y*z + z , x z + y z + 3y z + 3y*z +
-----------------------------------------------------------------------
5
z )
o49 : Ideal of R
i50 : gens gb L1
o50 = | xz3+yz3+z4 xy3+y4+y3z x3y+y4-x3z+2y3z-2yz3-z4 x4-y4+2x3z-2y3z+2yz3+z4
-----------------------------------------------------------------------
x3z2+y3z2+3y2z3+3yz4+z5 |
1 5
o50 : Matrix R <--- R
i51 : (gens L1) % F
o51 = 0
1 5
o51 : Matrix R <--- R
i52 : J = ideal ((gens L1)//F)
3 3 2 2 3 2 2 2 2 3 3 2
o52 = ideal (z , y , x y - x*y + y - x z + y z + x*z - y*z - z , x - x y
-----------------------------------------------------------------------
2 3 2 2 2 2 3 2 2 2 2 2 3
+ x*y - y + x z - y z - x*z + y*z + z , x z - x*y*z + y z - x*z
-----------------------------------------------------------------------
3 4
+ 2y*z + z )
o52 : Ideal of R
i53 : mingens J
o53 = | z3 y3 x2y-xy2-x2z+y2z+xz2-yz2 x3 x2z2-xyz2+y2z2 |
1 5
o53 : Matrix R <--- R
i54 : betti oo
0 1
o54 = total: 1 5
0: 1 .
1: . .
2: . 4
3: . 1
o54 : BettiTally
i55 : R = KK[x,y,z]
o55 = R
o55 : PolynomialRing
i56 : I = ideal(x^3,y^3,z^3)
3 3 3
o56 = ideal (x , y , z )
o56 : Ideal of R
i57 : F = x+y+z
o57 = x + y + z
o57 : R
i58 : J = I : F
3 3 2 2 2 2 2 2 3 2 2 2
o58 = ideal (z , y , x y - x*y - x z + y z + x*z - y*z , x , x z - x*y*z
-----------------------------------------------------------------------
2 2
+ y z )
o58 : Ideal of R
i59 : betti J
0 1
o59 = total: 1 5
0: 1 .
1: . .
2: . 4
3: . 1
o59 : BettiTally
i60 : transpose gens J
o60 = {-3} | z3 |
{-3} | y3 |
{-3} | x2y-xy2-x2z+y2z+xz2-yz2 |
{-3} | x3 |
{-4} | x2z2-xyz2+y2z2 |
5 1
o60 : Matrix R <--- R
i61 : transpose gens gb J
o61 = {-3} | z3 |
{-3} | y3 |
{-3} | x2y-xy2-x2z+y2z+xz2-yz2 |
{-3} | x3 |
{-4} | x2z2-xyz2+y2z2 |
5 1
o61 : Matrix R <--- R
i62 : R = KK[t,a..f]
o62 = R
o62 : PolynomialRing
i63 : I = ideal(a*b*c-d*e*f, a^2*b-c^2*d, a*f^2-d*b*c)
2 2 2
o63 = ideal (a*b*c - d*e*f, a b - c d, - b*c*d + a*f )
o63 : Ideal of R
i64 : F = a*b*c*d*e*f
o64 = a*b*c*d*e*f
o64 : R
i65 : J = eliminate(I + ideal(t*F-1), t)
2 2 2 2 3
o65 = ideal (d e - a f, b*d*e - c f, b*c*d - a*f , c - a*e*f, a*b*c - d*e*f,
-----------------------------------------------------------------------
2 2 2 2 3 4 2 2 3
a*b - c*f , a b - c d, b d - f , b c - e*f )
o65 : Ideal of R
i66 : transpose gens J
o66 = {-3} | d2e-a2f |
{-3} | bde-c2f |
{-3} | bcd-af2 |
{-3} | c3-aef |
{-3} | abc-def |
{-3} | ab2-cf2 |
{-3} | a2b-c2d |
{-4} | b3d-f4 |
{-4} | b2c2-ef3 |
9 1
o66 : Matrix R <--- R
i67 : R = KK[a..f]
o67 = R
o67 : PolynomialRing
i68 : I = substitute(I,R)
2 2 2
o68 = ideal (a*b*c - d*e*f, a b - c d, - b*c*d + a*f )
o68 : Ideal of R
i69 : F = product gens R
o69 = a*b*c*d*e*f
o69 : R
i70 : J' = saturate(I,F)
2 2 2 2 3
o70 = ideal (d e - a f, b*d*e - c f, b*c*d - a*f , c - a*e*f, a*b*c - d*e*f,
-----------------------------------------------------------------------
2 2 2 2 3 4 2 2 3
a*b - c*f , a b - c d, b d - f , b c - e*f )
o70 : Ideal of R
i71 : transpose gens J'
o71 = {-3} | d2e-a2f |
{-3} | bde-c2f |
{-3} | bcd-af2 |
{-3} | c3-aef |
{-3} | abc-def |
{-3} | ab2-cf2 |
{-3} | a2b-c2d |
{-4} | b3d-f4 |
{-4} | b2c2-ef3 |
9 1
o71 : Matrix R <--- R
i72 :
