If M is an ideal or module over a ring R, and F→M is a surjection from a free module, then reesAlgebra(M) returns the ring Sym(F)/J, where J = reesIdeal(M).
i1 : S = QQ[x_0..x_4] o1 = S o1 : PolynomialRing |
i2 : i = monomialCurveIdeal(S,{2,3,5,6}) 2 3 2 2 2 2 o2 = ideal (x x - x x , x - x x , x x - x x , x - x x , x x - x x , x x 2 3 1 4 2 0 4 1 2 0 3 3 2 4 1 3 0 4 0 3 ------------------------------------------------------------------------ 2 2 3 2 - x x , x x - x x x , x - x x ) 1 4 1 3 0 2 4 1 0 4 o2 : Ideal of S |
i3 : time I = reesIdeal i; -- used 0.097253 seconds o3 : Ideal of S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 |
i4 : reesIdeal(i, Variable=>v) o4 = ideal (x v - x v + x v , x v - x v - v , x v - x v + x v , x v - 2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 ------------------------------------------------------------------------ 2 x v - x v , x v - x v - x v , x v + x v - x v , x x v + x v - 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 ------------------------------------------------------------------------ 2 2 x v , x v - x v + x v + x v , x v + x v - x v , x x v - x x v - 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 ------------------------------------------------------------------------ 2 2 x v + x v , x x v - x x v - x v + x v - x v , x v - x v - x v + 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 ------------------------------------------------------------------------ 2 2 x v , x v v + v v - v v , x x v - v - x v v + v v , x x v v - 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 ------------------------------------------------------------------------ 2 x v v + v - v v ) 4 1 4 4 3 6 o4 : Ideal of S[v , v , v , v , v , v , v , v ] 0 1 2 3 4 5 6 7 |
i5 : time I=reesIdeal(i,i_0); -- used 0.60532 seconds o5 : Ideal of S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 |
i6 : time (J=symmetricKernel gens i); -- used 0. seconds o6 : Ideal of S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 |
i7 : isLinearType(i,i_0) o7 = false |
i8 : isLinearType i o8 = false |
i9 : reesAlgebra (i,i_0) S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 o9 = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 (x w - x w + x w , x w - x w - w , x w - x w + x w , x w - x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w - x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w - x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w + x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w + w - w w ) 2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4 4 3 6 o9 : QuotientRing |
i10 : trim ideal normalCone (i, i_0) 2 3 2 2 2 o10 = ideal (x x - x x , x - x x , x x - x x , x - x x , x x - x x , 2 3 1 4 2 0 4 1 2 0 3 3 2 4 1 3 0 4 ----------------------------------------------------------------------- 2 3 2 x x - x x x , x - x x ) 1 3 0 2 4 1 0 4 S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 o10 : Ideal of ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 (x w - x w + x w , x w - x w - w , x w - x w + x w , x w - x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w - x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w - x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w + x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w + w - w w ) 2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4 4 3 6 |
i11 : trim ideal associatedGradedRing (i,i_0) 2 3 2 2 2 o11 = ideal (x x - x x , x - x x , x x - x x , x - x x , x x - x x , 2 3 1 4 2 0 4 1 2 0 3 3 2 4 1 3 0 4 ----------------------------------------------------------------------- 2 3 2 x x - x x x , x - x x ) 1 3 0 2 4 1 0 4 S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 o11 : Ideal of ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 2 2 2 2 2 2 2 2 (x w - x w + x w , x w - x w - w , x w - x w + x w , x w - x w - x w , x w - x w - x w , x w + x w - x w , x x w + x w - x w , x w - x w + x w + x w , x w + x w - x w , x x w - x x w - x w + x w , x x w - x x w - x w + x w - x w , x w - x w - x w + x w , x w w + w w - w w , x x w - w - x w w + w w , x x w w - x w w + w - w w ) 2 0 3 1 4 2 1 0 3 2 5 0 0 1 1 2 2 0 4 1 5 4 7 0 3 3 5 4 6 4 2 1 3 3 4 0 4 2 1 6 3 7 3 2 2 4 3 5 4 6 1 2 0 6 2 7 1 4 1 2 4 2 1 4 3 6 0 4 1 1 3 2 1 5 2 6 4 7 3 0 4 1 2 3 4 4 4 2 5 4 6 3 7 1 4 2 6 4 1 7 4 7 3 4 0 2 4 1 4 4 3 6 |
i12 : trim specialFiberIdeal (i,i_0) 2 2 o12 = ideal (x , x , x , x , x , w , w - w w , w w - w w , w - w w ) 4 3 2 1 0 5 6 4 7 4 6 3 7 4 3 6 o12 : Ideal of S[w , w , w , w , w , w , w , w ] 0 1 2 3 4 5 6 7 |