+ M2 --no-readline --print-width 114 Macaulay2, version 1.4 with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone i1 : R := QQ[x,y,z] o1 = QQ[x, y, z] o1 : PolynomialRing i2 : I := ideal(x, x-y^2) 2 o2 = ideal (x, - y + x) o2 : Ideal of QQ[x, y, z] i3 : J := trim I 2 o3 = ideal (x, y ) o3 : Ideal of QQ[x, y, z] i4 : J == I o4 = true i5 : K := ideal(x^2+y^2+z^2) 2 2 2 o5 = ideal(x + y + z ) o5 : Ideal of QQ[x, y, z] i6 : L := trim (J + K) 2 2 o6 = ideal (x, z , y ) o6 : Ideal of QQ[x, y, z] i7 : J + K 2 2 2 2 o7 = ideal (x, y , x + y + z ) o7 : Ideal of QQ[x, y, z] i8 : radical L o8 = monomialIdeal (x, y, z) o8 : MonomialIdeal of QQ[x, y, z] i9 : isPrime (J+K) o9 = false i10 : isPrime (radical (J+K)) o10 = true i11 : isPrimary (J+K) o11 = true i12 : I 2 o12 = ideal (x, - y + x) o12 : Ideal of QQ[x, y, z] i13 : J 2 o13 = ideal (x, y ) o13 : Ideal of QQ[x, y, z] i14 : isPrimary J o14 = true i15 : isPrimary (ideal(x^2, x*y)) o15 = false i16 : L 2 2 o16 = ideal (x, z , y ) o16 : Ideal of QQ[x, y, z] i17 : M := ideal(y^2-x^3-x-1, x^2-x^3*z+5*z^7) 3 2 7 3 2 o17 = ideal (- x + y - x - 1, 5z - x z + x ) o17 : Ideal of QQ[x, y, z] i18 : intersect(L, M) 4 2 2 3 2 2 2 2 2 3 2 4 2 2 6 7 3 2 5 4 o18 = ideal (x - x*y + x + x, x z - y z + x*z + z , x y - y + x*y + y , - x z - 5z + x y z + x - x z - ------------------------------------------------------------------------------------------------------------ 2 2 3 x y + x ) o18 : Ideal of QQ[x, y, z] i19 : trim intersect(L, M) 4 2 2 3 2 2 2 2 2 3 2 4 2 2 7 3 2 o19 = ideal (x - x*y + x + x, x z - y z + x*z + z , x y - y + x*y + y , 5z - x z + x ) o19 : Ideal of QQ[x, y, z] i20 : intersect(L, M) == (L*M) o20 = true i21 : L 2 2 o21 = ideal (x, z , y ) o21 : Ideal of QQ[x, y, z] i22 : ass L o22 = {ideal (z, y, x)} o22 : List i23 : I := (ideal(x,y))^2*(ideal(y,z))^3 stdio:23:1:(3): warning: local declaration of I shields variable with same name 2 3 2 2 2 2 2 3 4 3 2 2 3 5 4 3 2 2 3 o23 = ideal (x y , x y z, x y*z , x z , x*y , x*y z, x*y z , x*y*z , y , y z, y z , y z ) o23 : Ideal of QQ[x, y, z] i24 : ass I o24 = {ideal (y, x), ideal (z, y), ideal (z, y, x)} o24 : List i25 : ass (ideal(x,y))^2 o25 = {ideal (y, x)} o25 : List i26 : primaryDecomposition I 2 2 3 2 2 3 2 4 5 3 4 2 2 3 2 3 o26 = {ideal (x , x*y, y ), ideal (y , y z, y*z , z ), ideal (x , x*y , y , x*y z, y z, x*y z , y z , z )} o26 : List i27 : intersect((ideal(x,y))^2, (ideal(y,z))^3) == I i27 : primaryDecomposition intersect((ideal(x,y)^2, (ideal(y,z))^3) primaryDecomposition intersect((ideal(x,y)^2, (ideal(y,z))^3) stdio:30:1:(3): error: missing semicolon or comma on previous line? i27 : primaryDecomposition intersect((ideal(x,y)^2, (ideal(y,z))^3)) primaryDecomposition intersect((ideal(x,y)^2, (ideal(y,z))^3) i27 : primaryDecomposition intersect((ideal(x,y))^2, (ideal(y,z))^3) stdio:32:43:(3):[1]: error: no method for binary operator ^ applied to objects: -- (x, y) (of class Sequence) -- ^ 2 (of class ZZ) i28 : primaryDecomposition intersect((ideal(x,y))^2, (ideal(y,z))^3) 2 2 3 2 2 3 o28 = {ideal (x , x*y, y ), ideal (y , y z, y*z , z )} o28 : List i29 : intersect((ideal(x,y))^2, (ideal(y,z))^3) == I o29 = false i30 : primaryDecomposition I 2 2 3 2 2 3 2 4 5 3 4 2 2 3 2 3 o30 = {ideal (x , x*y, y ), ideal (y , y z, y*z , z ), ideal (x , x*y , y , x*y z, y z, x*y z , y z , z )} o30 : List i31 : exit Process M2 finished + M2 --no-readline --print-width 114 Macaulay2, version 1.4 with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone i1 : R := ZZ/17[x,y,z] ZZ o1 = --[x, y, z] 17 o1 : PolynomialRing i2 : S := R/ideal(x*y-z^2) ZZ --[x, y, z] 17 o2 = ----------- 2 x*y - z o2 : QuotientRing i3 : I := ideal(x_S, z_S) o3 = ideal (x, z) ZZ --[x, y, z] 17 o3 : Ideal of ----------- 2 x*y - z i4 : ideal(x, z) o4 = ideal (x, z) ZZ o4 : Ideal of --[x, y, z] 17 i5 : dim R o5 = 3 i6 : dim S o6 = 2 i7 : isPrime ideal(x*y - z^2) o7 = true i8 : J := ideal(x, z) o8 = ideal (x, z) ZZ o8 : Ideal of --[x, y, z] 17 i9 : dim (R/J) o9 = 1 i10 : dim (S/J) stdio:10:7:(3): error: expected ideal of the same ring i11 : dim (S/I) o11 = 1 i12 : isPrime I o12 = true i13 : isPrimary I^2 o13 = false i14 : ass I^2 o14 = {ideal (z, x), ideal (z, y, x)} o14 : List i15 : IK o15 = IK o15 : Symbol i16 : I o16 = ideal (x, z) ZZ --[x, y, z] 17 o16 : Ideal of ----------- 2 x*y - z i17 : primaryDecomposition I^2 2 o17 = {ideal(x), ideal (y, x*z, x )} o17 : List i18 : exit Process M2 finished