-- -*- M2-comint -*- {* hash: -139854778 *}
i1 : R = QQ[a..h]
o1 = R
o1 : PolynomialRing
i2 : rows = {0,1,2}
o2 = {0, 1, 2}
o2 : List
i3 : cols = {0,3}
o3 = {0, 3}
o3 : List
i4 : result = map(R^3, 2, (i,j) -> R_(rows_i + cols_j))
o4 = | a d |
| b e |
| c f |
3 2
o4 : Matrix R <--- R
i5 : R = ZZ/101[a..d];
i6 : m = matrix{{a^2+a^2*c+a*b+3*d}}
o6 = | a2c+a2+ab+3d |
1 1
o6 : Matrix R <--- R
i7 : result = coefficients(m, Variables => {a})
o7 = (| a2 a 1 |, {2} | c+1 |)
{1} | b |
{0} | 3d |
o7 : Sequence
i8 : result_0
o8 = | a2 a 1 |
1 3
o8 : Matrix R <--- R
i9 : result_1
o9 = {2} | c+1 |
{1} | b |
{0} | 3d |
3 1
o9 : Matrix R <--- R
i10 : R = QQ[a,b,Degrees=>{{1,0},{1,-1}}];
i11 : m = matrix{{a*b, b^2}}
o11 = | ab b2 |
1 2
o11 : Matrix R <--- R
i12 : (degrees source m)_0
o12 = {2, -1}
o12 : List
i13 : R = ZZ/101[a..d]
o13 = R
o13 : PolynomialRing
i14 : m = matrix{{a,b},{c,d}}
o14 = | a b |
| c d |
2 2
o14 : Matrix R <--- R
i15 : copym = map(target m, source m, entries m)
o15 = | a b |
| c d |
2 2
o15 : Matrix R <--- R
i16 : R = ZZ[a..d];
i17 : m = matrix{{a^2,b^3,c^4,d^5}}
o17 = | a2 b3 c4 d5 |
1 4
o17 : Matrix R <--- R
i18 : map(R^(numgens source m), source m,
(i,j) -> if i === j then m_(0,i) else 0)
o18 = | a2 0 0 0 |
| 0 b3 0 0 |
| 0 0 c4 0 |
| 0 0 0 d5 |
4 4
o18 : Matrix R <--- R
i19 : R = ZZ[a..d];
i20 : m = matrix{{a,b^2},{c^2,d^3}}
o20 = | a b2 |
| c2 d3 |
2 2
o20 : Matrix R <--- R
i21 : betti m
0 1
o21 = total: 2 2
0: 2 .
1: . 1
2: . 1
o21 : BettiTally
i22 : n = m ** R^{-1}
o22 = {1} | a b2 |
{1} | c2 d3 |
2 2
o22 : Matrix R <--- R
i23 : betti n
0 1
o23 = total: 2 2
1: 2 .
2: . 1
3: . 1
o23 : BettiTally
i24 : R = QQ[a..d]
o24 = R
o24 : PolynomialRing
i25 : S = QQ[s,t]
o25 = S
o25 : PolynomialRing
i26 : m = matrix{{a^2-d, b*c}}
o26 = | a2-d bc |
1 2
o26 : Matrix R <--- R
i27 : f = matrix{{s^4,s^3*t,s*t^3,t^4}}
o27 = | s4 s3t st3 t4 |
1 4
o27 : Matrix S <--- S
i28 : substitute(m,f)
o28 = | s8-t4 s4t4 |
1 2
o28 : Matrix S <--- S
i29 : F = map(R,R,{b,c,d,a})
o29 = map(R,R,{b, c, d, a})
o29 : RingMap R <--- R
i30 : m + F m + F F m + F F F m
o30 = | a2+b2+c2+d2-a-b-c-d ab+bc+ad+cd |
1 2
o30 : Matrix R <--- R
i31 : substitute(m, {a=>1, b=>3})
o31 = | -d+1 3c |
1 2
o31 : Matrix R <--- R
i32 : R = ZZ[s,t]
o32 = R
o32 : PolynomialRing
i33 : m = s^2+t^2
2 2
o33 = s + t
o33 : R
i34 : S = R[a..d]
o34 = S
o34 : PolynomialRing
i35 : substitute(m,S)
2 2
o35 = s + t
o35 : S
i36 : R = ZZ[a..d]
o36 = R
o36 : PolynomialRing
i37 : f = matrix{{a^2-b*c,3*b*c^4-1}}
o37 = | a2-bc 3bc4-1 |
1 2
o37 : Matrix R <--- R
i38 : J = ideal f
2 4
o38 = ideal (a - b*c, 3b*c - 1)
o38 : Ideal of R
i39 : generators J
o39 = | a2-bc 3bc4-1 |
1 2
o39 : Matrix R <--- R
i40 : image f
o40 = image | a2-bc 3bc4-1 |
1
o40 : R-module, submodule of R
i41 : cokernel f
o41 = cokernel | a2-bc 3bc4-1 |
1
o41 : R-module, quotient of R
i42 : id_(R^4)
o42 = | 1 0 0 0 |
| 0 1 0 0 |
| 0 0 1 0 |
| 0 0 0 1 |
4 4
o42 : Matrix R <--- R
i43 : myanswer = 2*(numgens R) - 1
o43 = 7
i44 : R = ZZ/31991[a..d]
o44 = R
o44 : PolynomialRing
i45 : a
o45 = a
o45 : R
i46 : a = 43
o46 = 43
i47 : a
o47 = 43
i48 : use R
o48 = R
o48 : PolynomialRing
i49 : a
o49 = a
o49 : R
i50 : I = ideal(a^2-b,c-1,d^2-a*b)
2 2
o50 = ideal (a - b, c - 1, - a*b + d )
o50 : Ideal of R
i51 : J = ideal(a*b-1, c*d-2)
o51 = ideal (a*b - 1, c*d - 2)
o51 : Ideal of R
i52 : intersect(I,J)
2 3
o52 = ideal (c d - c*d - 2c + 2, a*b*c - a*b - c + 1, - 10664c*d +
-----------------------------------------------------------------------
2 2 2
10664a*b*d + 10663a*b + 10664c*d - 10663d - 10664d, - 10664c d +
-----------------------------------------------------------------------
2 2
10664a*b*c + 10663c d + 10664c*d - 10664a*b + 10665c*d + c + 10663d -
-----------------------------------------------------------------------
2 2 3 2 2 2
1, - 10664b c*d + 10664a*b + 10663b c*d + 10664a*c*d - 10665a b -
-----------------------------------------------------------------------
2 2 2 2 2
10663b d - 10663a*c*d + b + 10663a*d, - 10664a*b*c*d + 10664a b +
-----------------------------------------------------------------------
2
10663a*b*c*d - 10663a*b*d + 10664c*d - 10664a*b - 10663c*d + 10663d, -
-----------------------------------------------------------------------
2 2 3 2 2 2
10664a c*d + 10664a b + 10663a c*d + 10664b*c*d - 10664a*b -
-----------------------------------------------------------------------
2 2
10663a d - 10663b*c*d + a + 10663b*d - b)
o52 : Ideal of R
i53 :
