-- -*- M2-comint -*- {* hash: -743212642 *}
i1 : R = ZZ/10007[a,b];
i2 : f = (2*a+3)^4 + 5
4 3 2
o2 = 16a + 96a + 216a + 216a + 86
o2 : R
i3 : g = (2*a+b+1)^3
3 2 2 3 2 2
o3 = 8a + 12a b + 6a*b + b + 12a + 12a*b + 3b + 6a + 3b + 1
o3 : R
i4 : size f, size g
o4 = (5, 10)
o4 : Sequence
i5 : degree f
o5 = {4}
o5 : List
i6 : degree g
o6 = {3}
o6 : List
i7 : terms g
3 2 2 3 2 2
o7 = {8a , 12a b, 6a*b , b , 12a , 12a*b, 3b , 6a, 3b, 1}
o7 : List
i8 : select(terms g, i -> degree i == {2})
2 2
o8 = {12a , 12a*b, 3b }
o8 : List
i9 : sum oo
2 2
o9 = 12a + 12a*b + 3b
o9 : R
i10 : part(0,g)
o10 = 1
o10 : R
i11 : part(1,g)
o11 = 6a + 3b
o11 : R
i12 : part(2,g)
2 2
o12 = 12a + 12a*b + 3b
o12 : R
i13 : part(3,g)
3 2 2 3
o13 = 8a + 12a b + 6a*b + b
o13 : R
i14 : toString f
o14 = 16*a^4+96*a^3+216*a^2+216*a+86
i15 : toString g
o15 = 8*a^3+12*a^2*b+6*a*b^2+b^3+12*a^2+12*a*b+3*b^2+6*a+3*b+1
i16 : quot = f//g
o16 = 2a - 3b + 9
o16 : R
i17 : rem = f%g
2 2 3 4 2 2 2 2
o17 = 24a b + 16a*b + 3b - 96a b - 24a*b + 96a - 96a*b - 18b + 160a -
-----------------------------------------------------------------------
24b + 77
o17 : R
i18 : f == quot * g + rem
o18 = true
i19 : homogenize(f,b)
4 3 2 2 3 4
o19 = 16a + 96a b + 216a b + 216a*b + 86b
o19 : R
i20 : ring f
o20 = R
o20 : PolynomialRing
i21 : ring f === ring g
o21 = true
i22 : part(1,f)
o22 = 216a
o22 : R
i23 : f_a
o23 = 216
ZZ
o23 : -----
10007
i24 : g_(a*b)
o24 = 12
ZZ
o24 : -----
10007
i25 : leadTerm g
3
o25 = 8a
o25 : R
i26 : leadCoefficient g
o26 = 8
ZZ
o26 : -----
10007
i27 : leadMonomial g
3
o27 = a
o27 : R
i28 : exponents leadMonomial g
o28 = {{3, 0}}
o28 : List
i29 : coefficients f
o29 = (| a4 a3 a2 a 1 |, {4} | 16 |)
{3} | 96 |
{2} | 216 |
{1} | 216 |
{0} | 86 |
o29 : Sequence
i30 : coefficients g
o30 = (| a3 a2b ab2 b3 a2 ab b2 a b 1 |, {3} | 8 |)
{3} | 12 |
{3} | 6 |
{3} | 1 |
{2} | 12 |
{2} | 12 |
{2} | 3 |
{1} | 6 |
{1} | 3 |
{0} | 1 |
o30 : Sequence
i31 : exponents f
o31 = {{4, 0}, {3, 0}, {2, 0}, {1, 0}, {0, 0}}
o31 : List
i32 : exponents g
o32 = {{3, 0}, {2, 1}, {1, 2}, {0, 3}, {2, 0}, {1, 1}, {0, 2}, {1, 0}, {0,
-----------------------------------------------------------------------
1}, {0, 0}}
o32 : List
i33 : listForm f
o33 = {({4, 0}, 16), ({3, 0}, 96), ({2, 0}, 216), ({1, 0}, 216), ({0, 0},
-----------------------------------------------------------------------
86)}
o33 : List
i34 : S = listForm g
o34 = {({3, 0}, 8), ({2, 1}, 12), ({1, 2}, 6), ({0, 3}, 1), ({2, 0}, 12),
-----------------------------------------------------------------------
({1, 1}, 12), ({0, 2}, 3), ({1, 0}, 6), ({0, 1}, 3), ({0, 0}, 1)}
o34 : List
i35 : S / print;
({3, 0}, 8)
({2, 1}, 12)
({1, 2}, 6)
({0, 3}, 1)
({2, 0}, 12)
({1, 1}, 12)
({0, 2}, 3)
({1, 0}, 6)
({0, 1}, 3)
({0, 0}, 1)
i36 : S = standardForm f
o36 = HashTable{HashTable{} => 86 }
HashTable{0 => 1} => 216
HashTable{0 => 2} => 216
HashTable{0 => 3} => 96
HashTable{0 => 4} => 16
o36 : HashTable
i37 : standardForm g
o37 = HashTable{HashTable{} => 1 }
HashTable{0 => 1} => 6
HashTable{0 => 1} => 12
1 => 1
HashTable{0 => 1} => 6
1 => 2
HashTable{0 => 2} => 12
HashTable{0 => 2} => 12
1 => 1
HashTable{0 => 3} => 8
HashTable{1 => 1} => 3
HashTable{1 => 2} => 3
HashTable{1 => 3} => 1
o37 : HashTable
i38 : S#(new HashTable from {0 => 2})
o38 = 216
ZZ
o38 : -----
10007
i39 : f < g
o39 = false
i40 : sort {b^2-1,a*b,a+1,a,b}
2
o40 = {b, a, a + 1, b - 1, a*b}
o40 : List
i41 : f ? g
o41 = >
o41 : Keyword
i42 :
