-- -*- M2-comint -*- {* hash: -430286454 *}
i1 : KK = ZZ/31991
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[a..d]
o5 = R
o5 : PolynomialRing
i6 : I = monomialCurveIdeal(R,{1,3,4})
3 2 2 2 3 2
o6 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o6 : Ideal of R
i7 : codim I
o7 = 2
i8 : dim I
o8 = 2
i9 : codim (R^1/(I*R^1))
o9 = 2
i10 : M = coker gens I
o10 = cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c |
1
o10 : R-module, quotient of R
i11 : codim M
o11 = 2
i12 : dim M
o12 = 2
i13 : degree I
o13 = 4
i14 : degree M
o14 = 4
i15 : hilbertPolynomial M
o15 = - 3*P + 4*P
0 1
o15 : ProjectiveHilbertPolynomial
i16 : hilbertSeries M
2 3 4 5
1 - T - 3T + 4T - T
o16 = -----------------------
4
(1 - T)
o16 : Expression of class Divide
i17 : Mres = res M
1 4 4 1
o17 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o17 : ChainComplex
i18 : betti Mres
0 1 2 3
o18 = total: 1 4 4 1
0: 1 . . .
1: . 1 . .
2: . 3 4 1
o18 : BettiTally
i19 : R = KK[x,y,z]
o19 = R
o19 : PolynomialRing
i20 : F = random(R^1, R^{-2,-3})
o20 = | 305x2-8319xy+14216y2+4825xz+1360yz-7636z2
-----------------------------------------------------------------------
-11783x3+4017x2y+1915xy2-13519y3-6295x2z-14966xyz-3634y2z-9977xz2-
-----------------------------------------------------------------------
8554yz2-1894z3 |
1 2
o20 : Matrix R <--- R
i21 : GB = gens gb F
o21 = | x2-15341xy+15675y2+12078xz+4200yz-10409z2
-----------------------------------------------------------------------
xy2+5163y3+7367xyz-2253y2z-7818xz2-10970yz2-15144z3
-----------------------------------------------------------------------
y4+2900y3z-13269xyz2-9273y2z2+11675xz3-3549yz3+1498z4 |
1 3
o21 : Matrix R <--- R
i22 : LT = leadTerm gens gb F
o22 = | x2 xy2 y4 |
1 3
o22 : Matrix R <--- R
i23 : betti LT
0 1
o23 = total: 1 3
0: 1 .
1: . 1
2: . 1
3: . 1
o23 : BettiTally
i24 : R = KK[x,y,z, MonomialOrder => Lex]
o24 = R
o24 : PolynomialRing
i25 : F = random(R^1, R^{-2,-3})
o25 = | -4743x2-15027xy-2999xz-1334y2-12991yz+7321z2
-----------------------------------------------------------------------
-1568x3-9457x2y-6588x2z+376xy2-12945xyz-6735xz2-2368y3-2846y2z+5916yz2+
-----------------------------------------------------------------------
4396z3 |
1 2
o25 : Matrix R <--- R
i26 : GB = gens gb F
o26 = | y6+6781y5z+9886y4z2-5922y3z3+15586y2z4+3142yz5+3816z6
-----------------------------------------------------------------------
xz4+1736y5+13680y4z-3939y3z2-15144y2z3-10905yz4+1763z5
-----------------------------------------------------------------------
xyz2+6036xz3-15150y4-10154y3z-14712y2z2+619yz3-13974z4
-----------------------------------------------------------------------
xy2-623xyz+14807xz2-5117y3-8439y2z+12868yz2-10042z3
-----------------------------------------------------------------------
x2-9669xy-13017xz-9193y2+15489yz-1708z2 |
1 5
o26 : Matrix R <--- R
i27 : LT = leadTerm gens gb F
o27 = | y6 xz4 xyz2 xy2 x2 |
1 5
o27 : Matrix R <--- R
i28 : betti LT
0 1
o28 = total: 1 5
0: 1 .
1: . 1
2: . 1
3: . 1
4: . 1
5: . 1
o28 : BettiTally
i29 : R = KK[a..i]
o29 = R
o29 : PolynomialRing
i30 : M = genericMatrix(R,a,3,3)
o30 = | a d g |
| b e h |
| c f i |
3 3
o30 : Matrix R <--- R
i31 : N = M^3
o31 = | 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
o31 : Matrix R <--- R
i32 : I = flatten N
o32 = | 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
o32 : Matrix R <--- R
i33 : Tr = trace M
o33 = a + e + i
o33 : R
i34 : Tr //I -- the quotient, which is 0
o34 = 0
9 1
o34 : Matrix R <--- R
i35 : Tr % I -- the remainder, which is Tr again
o35 = a + e + i
o35 : R
i36 : Tr^2 % I
2 2 2
o36 = a + 2a*e + e + 2a*i + 2e*i + i
o36 : R
i37 : Tr^3 % I
2 2 3 2
o37 = 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
o37 : R
i38 : Tr^4 % I
2 2 2 3 4
o38 = 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
o38 : R
i39 : Tr^5 % I
2 2 2 3 4 2 2
o39 = 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
o39 : R
i40 : Tr^6 % I
2 2 2 2 2 3 2 4 2 2 2 2 2
o40 = 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
o40 : R
i41 : Tr^7 % I
o41 = 0
o41 : R
i42 : Tr^6 // I
o42 = {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
o42 : Matrix R <--- R
i43 : Tr^6 == I * (Tr^6 // I) + (Tr^6 % I)
o43 = true
i44 : x = global x
o44 = x
o44 : Symbol
i45 : R = KK[x_0..x_3]
o45 = R
o45 : PolynomialRing
i46 : M = map(R^2, 3, (i,j)->x_(i+j))
o46 = | x_0 x_1 x_2 |
| x_1 x_2 x_3 |
2 3
o46 : Matrix R <--- R
i47 : I = gens minors(2,M)
o47 = | -x_1^2+x_0x_2 -x_1x_2+x_0x_3 -x_2^2+x_1x_3 |
1 3
o47 : Matrix R <--- R
i48 : pideal = ideal(x_0+x_3, x_1, x_2)
o48 = ideal (x + x , x , x )
0 3 1 2
o48 : Ideal of R
i49 : y = global y
o49 = y
o49 : Symbol
i50 : S = KK[y_0..y_3,MonomialOrder=> Eliminate 1]
o50 = S
o50 : PolynomialRing
i51 : I1 = substitute(I, matrix{{y_0,y_1,y_2,y_3-y_0}})
o51 = | y_0y_2-y_1^2 -y_0^2+y_0y_3-y_1y_2 -y_0y_1-y_2^2+y_1y_3 |
1 3
o51 : Matrix S <--- S
i52 : J = selectInSubring(1,gens gb I1)
o52 = | y_1^3+y_2^3-y_1y_2y_3 |
1 1
o52 : Matrix S <--- S
i53 : S1 = KK[y_1..y_3]
o53 = S1
o53 : PolynomialRing
i54 : J1 = substitute(J, S1)
o54 = | y_1^3+y_2^3-y_1y_2y_3 |
1 1
o54 : Matrix S1 <--- S1
i55 : Rbar = R/(ideal I)
o55 = Rbar
o55 : QuotientRing
i56 : f = map(Rbar, S1, matrix(Rbar,{{x_0+x_3, x_1,x_2}}))
o56 = map(Rbar,S1,{x + x , x , x })
0 3 1 2
o56 : RingMap Rbar <--- S1
i57 : J1 = ker f
3 3
o57 = ideal(y - y y y + y )
2 1 2 3 3
o57 : Ideal of S1
i58 : R = KK[a,b,c,d]
o58 = R
o58 : PolynomialRing
i59 : I1 = ideal(d*b-a^2, d^2*c-a^3)
2 3 2
o59 = ideal (- a + b*d, - a + c*d )
o59 : Ideal of R
i60 : I1aug = (gens I1) | matrix{{d}}
o60 = | -a2+bd -a3+cd2 d |
1 3
o60 : Matrix R <--- R
i61 : augrelations = gens ker I1aug
o61 = {2} | -a d |
{3} | 1 0 |
{1} | ab-cd a2-bd |
3 2
o61 : Matrix R <--- R
i62 : I21 = submatrix(augrelations, {2}, {0,1})
o62 = {1} | ab-cd a2-bd |
1 2
o62 : Matrix R <--- R
i63 : I21 = ideal I21
2
o63 = ideal (a*b - c*d, a - b*d)
o63 : Ideal of R
i64 : I22 = I21 : d
2 2
o64 = ideal (b - a*c, a*b - c*d, a - b*d)
o64 : Ideal of R
i65 : I23 = I22 : d
2 2
o65 = ideal (b - a*c, a*b - c*d, a - b*d)
o65 : Ideal of R
i66 : (gens I23) % (gens I22)
o66 = 0
1 3
o66 : Matrix R <--- R
i67 : gens gb I1
o67 = | a2-bd abd-cd2 b2d2-acd2 |
1 3
o67 : Matrix R <--- R
i68 : I2 = divideByVariable(gens gb I1,d)
o68 = (| a2-bd ab-cd b2-ac |, 2)
o68 : Sequence
i69 : saturate(I1, d)
2 2
o69 = ideal (b - a*c, a*b - c*d, a - b*d)
o69 : Ideal of R
i70 :
