//This is an example from //A Module view of integral closures //in which there are more (10) basis elements in the integral closure // than in the ring (9). //The qthpoweralgorithm file must be loaded first to run this in MAGMA, //but the output, slightly edited, commented out below for thos who don't have MAGMA available to them. ////////////////////////////////////// F:=FiniteField(2); WT_MATRIX_S:=[[19,15,12,9,9],[12,9,9,9,0]]; S:=PolynomialRing(F,5,"weight", [19,15,12,9,9,12,9,9,9,0,1,1,1,0,0,1,1,0,0,0,1,0,0,0,0]); AssignNames(~S, ["x" cat &cat [ "_" cat IntegerToString(WT_MATRIX_S[i,j]) : i in [1..#WT_MATRIX_S]] : j in [1..#WT_MATRIX_S[1]]]); WT_MATRIX_icS,integral_closure_S,I,delta,phi,psi:= integral_closure(F,S, 2, [[19,15,12,9,9],[12,9,9,9,0]], [S|S.2^2+S.3*S.4*S.5,S.2*S.3+S.2+S.4*S.5*(S.4+S.5),S.3^2+S.3+S.2*(S.4+S.5)], S.1^3+S.1*(S.2+S.3)*(S.4+1)*S.5+(S.2*(S.4+S.5)+S.3*(S.4*S.5+1))*S.4^2*S.5); AssignNames(~integral_closure_S, ["y" cat &cat [ "_" cat IntegerToString(WT_MATRIX_icS[i,j]) : i in [1..#WT_MATRIX_icS]] : j in [1..#WT_MATRIX_icS[1]]]); "I="; for i in [1..#Basis(I)] do i,Basis(I)[i]; end for; "delta=",delta; "phi="; for i in [1..#WT_MATRIX_icS[1]] do i,integral_closure_S.i@phi; end for; "psi="; for i in [1..Rank(S)] do i,S.i@psi; end for; ///The computed Delta used in the qth power algorithm///////////////////////////////////////////////////////////////////////////// Delta=x_9_9^4*x_9_0^3+x_9_9^3*x_9_0^4+x_9_9^4*x_9_0^2+x_9_9^3*x_9_0^3+x_9_9^2*x_9_0^3+x_9_9*x_9_0^4+x_9_9^2*x_9_0^2+x_9_9*x_9_0^3 ///The basis from the qth power algorithm, multiplied by delta, the first element listed////////////////////////////////////////// 1 x_9_9^3*x_9_0^3+x_9_9^2*x_9_0^4+x_9_9^3*x_9_0^2+x_9_9*x_9_0^4+x_9_9^2*x_9_0^2+x_9_9*x_9_0^3 2 x_12_9*x_9_9^3*x_9_0^3+x_12_9*x_9_9^2*x_9_0^4+x_12_9*x_9_9^3*x_9_0^2+x_12_9*x_9_9*x_9_0^4+x_12_9*x_9_9^2*x_9_0^2+x_12_9*x_9_9*x_9_0^3 3 x_15_9*x_9_9^3*x_9_0^3+x_15_9*x_9_9^2*x_9_0^4+x_15_9*x_9_9^3*x_9_0^2+x_15_9*x_9_9*x_9_0^4+x_15_9*x_9_9^2*x_9_0^2+x_15_9*x_9_9*x_9_0^3 4 x_19_12*x_9_9^3*x_9_0^3+x_19_12*x_9_9^2*x_9_0^4+x_19_12*x_9_9^3*x_9_0^2+x_19_12*x_9_9*x_9_0^4+x_19_12*x_9_9^2*x_9_0^2+x_19_12*x_9_9*x_9_0^3 5 x_19_12^2*x_12_9*x_9_9*x_9_0^2+x_19_12^2*x_9_9^2*x_9_0^2+x_19_12^2*x_9_9*x_9_0^3+x_19_12^2*x_15_9*x_9_9*x_9_0+x_19_12^2*x_15_9*x_9_0^2+x_19_12^2*x_9_9*x_9_0^2+x_19_12^2*x_12_9*x_9_0+x_19_12^2*x_9_0 6 x_19_12^2*x_15_9*x_9_9^2*x_9_0+x_19_12^2*x_15_9*x_9_9*x_9_0^2+x_19_12^2*x_12_9*x_9_9^2*x_9_0+x_19_12^2*x_12_9*x_9_9*x_9_0^2+x_19_12^2*x_9_9^3*x_9_0+x_19_12^2*x_9_9*x_9_0^3+x_19_12^2*x_12_9*x_9_9*x_9_0+x_19_12^2*x_9_9^2*x_9_0+x_19_12^2*x_9_9*x_9_0^2+x_19_12^2*x_15_9*x_9_9+x_19_12^2*x_15_9*x_9_0+x_19_12^2*x_9_9*x_9_0+x_19_12^2*x_12_9+x_19_12^2 7 x_19_12^2*x_9_9^3*x_9_0^2+x_19_12^2*x_9_9^2*x_9_0^3+x_19_12*x_9_9^4*x_9_0^3+x_19_12*x_9_9^3*x_9_0^4+x_19_12^2*x_15_9*x_9_9^2*x_9_0+x_19_12^2*x_15_9*x_9_9*x_9_0^2+x_19_12^2*x_12_9*x_9_9^2*x_9_0+x_19_12^2*x_12_9*x_9_9*x_9_0^2+x_19_12*x_9_9^4*x_9_0^2+x_19_12*x_9_9^3*x_9_0^3+x_19_12^2*x_9_9^2*x_9_0+x_19_12^2*x_9_9*x_9_0^2+x_19_12*x_9_9^2*x_9_0^3+x_19_12*x_9_9*x_9_0^4+x_19_12*x_9_9^2*x_9_0^2+x_19_12*x_9_9*x_9_0^3+x_19_12*x_12_9*x_9_9^3*x_9_0^3+x_19_12*x_12_9*x_9_9^2*x_9_0^4+x_19_12*x_9_9^3*x_9_0^4+x_19_12*x_9_9^2*x_9_0^5+x_19_12*x_12_9*x_9_9^3*x_9_0^2+x_19_12*x_12_9*x_9_9*x_9_0^4+x_19_12*x_9_9^2*x_9_0^4+x_19_12*x_9_9*x_9_0^5+x_19_12*x_12_9*x_9_9^2*x_9_0^2+x_19_12*x_12_9*x_9_9*x_9_0^3+x_19_12*x_9_9^3*x_9_0^2+x_19_12*x_9_9^2*x_9_0^3+x_19_12*x_9_9^2*x_9_0^2+x_19_12*x_9_9*x_9_0^3 9 x_19_12*x_15_9*x_9_9^3*x_9_0^3+x_19_12*x_15_9*x_9_9^2*x_9_0^4+x_19_12*x_15_9*x_9_9^3*x_9_0^2+x_19_12*x_15_9*x_9_9*x_9_0^4+x_19_12*x_15_9*x_9_9^2*x_9_0^2+x_19_12*x_15_9*x_9_9*x_9_0^3 10 x_19_12*x_15_9*x_9_9^4*x_9_0^2+x_19_12*x_15_9*x_9_9^3*x_9_0^3+x_19_12*x_15_9*x_9_9^4*x_9_0+x_19_12*x_15_9*x_9_9^2*x_9_0^3+x_19_12*x_12_9*x_9_9^3*x_9_0^2+x_19_12*x_12_9*x_9_9^2*x_9_0^3+x_19_12*x_9_9^3*x_9_0^3+x_19_12*x_9_9^2*x_9_0^4+x_19_12*x_15_9*x_9_9^3*x_9_0+x_19_12*x_15_9*x_9_9^2*x_9_0^2+x_19_12*x_12_9*x_9_9^3*x_9_0+x_19_12*x_12_9*x_9_9*x_9_0^3+x_19_12*x_9_9^2*x_9_0^3+x_19_12*x_9_9*x_9_0^4+x_19_12*x_12_9*x_9_9^2*x_9_0+x_19_12*x_12_9*x_9_9*x_9_0^2+x_19_12*x_9_9^3*x_9_0+x_19_12*x_9_9^2*x_9_0^2 + x_19_12*x_9_9^2*x_9_0+x_19_12*x_9_9*x_9_0^2 ////The weight matrix for the integral closure using these functions///////////////////////////////// WT_MATRIX_T= [ [ 34, 34, 31, 29, 26, 23, 19, 15, 12, 9, 9 ], [ 30, 21, 21, 24, 24, 15, 12, 9, 9, 9, 0 ] ] ////The ideal of relations, using these functions written using their weights as names////////////////////////////////////////////// I= 1 y_34_30*y_9_0 +y_34_21*y_9_9+y_34_21*y_9_0+y_31_21+y_19_12 2 y_12_9^2 +y_15_9*y_9_9+y_15_9*y_9_0+y_12_9 3 y_15_9^2 +y_12_9*y_9_9*y_9_0 4 y_15_9*y_12_9 +y_15_9+y_9_9^2*y_9_0+y_9_9*y_9_0^2 5 y_19_12^2 +y_29_24*y_9_0+y_23_15 6 y_19_12*y_15_9 +y_34_21 7 y_19_12*y_12_9 +y_31_21 8 y_23_15^2 +y_23_15*y_9_9*y_9_0+y_23_15*y_9_0+y_19_12*y_9_9^2*y_9_0 9 y_23_15*y_19_12+y_19_12*y_9_9*y_9_0+y_19_12*y_9_0+y_15_9*y_9_9^2*y_9_0+y_12_9*y_9_9^2*y_9_0 10 y_23_15*y_15_9+y_29_24*y_9_0+y_26_24*y_9_0+y_23_15*y_9_0 11 y_23_15*y_12_9+y_26_24*y_9_0+y_23_15*y_9_0+y_23_15 12 y_26_24^2 +y_34_30*y_9_9^2+y_29_24*y_9_9^2+y_29_24*y_9_9*y_9_0+y_29_24*y_9_9+y_29_24*y_9_0+y_26_24*y_9_9^2+y_26_24*y_9_9*y_9_0+y_26_24*y_9_0+y_26_24+y_23_15*y_9_9^2+y_23_15+y_19_12*y_9_9^2*y_9_0 13 y_26_24*y_23_15+y_31_21*y_9_9^2+y_26_24*y_9_9*y_9_0+y_26_24*y_9_0+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9^2 14 y_26_24*y_19_12+y_31_21*y_9_9+y_31_21+y_19_12*y_9_9*y_9_0+y_19_12*y_9_9+y_19_12*y_9_0+y_19_12+y_15_9*y_9_9^3+y_12_9*y_9_9^2*y_9_0+y_9_9^4*y_9_0+y_9_9^3*y_9_0^2 15 y_26_24*y_15_9 +y_29_24*y_9_0+y_26_24*y_9_0+y_23_15*y_9_9^2+y_23_15*y_9_9*y_9_0+y_23_15*y_9_0 16 y_26_24*y_12_9 +y_29_24*y_9_9+y_29_24*y_9_0+y_26_24*y_9_9+y_23_15*y_9_9+y_23_15 17 y_29_24^2 +y_34_30*y_9_9^2+y_31_21*y_9_9^3+y_29_24*y_9_9^2+y_29_24*y_9_9*y_9_0+y_29_24*y_9_9+y_29_24*y_9_0+y_26_24*y_9_9^2*y_9_0+y_26_24*y_9_9^2+y_26_24*y_9_0+y_26_24+y_23_15*y_9_9^2*y_9_0+y_23_15*y_9_9+y_23_15*y_9_0+y_23_15 18 y_29_24*y_26_24+y_34_30*y_9_9^2+y_34_21*y_9_9^2+y_31_21*y_9_9^2+y_29_24*y_9_9^2+y_29_24*y_9_9+y_26_24*y_9_9^2+y_26_24*y_9_9*y_9_0+y_26_24*y_9_0+y_26_24+y_23_15*y_9_9^3+y_23_15*y_9_9^2*y_9_0+y_23_15*y_9_0+y_23_15+y_19_12*y_9_9^4+y_19_12*y_9_9^3*y_9_0+y_19_12*y_9_9^2 19 y_29_24*y_23_15+y_34_21*y_9_9^2+y_31_21*y_9_9^2+y_29_24*y_9_9*y_9_0+y_29_24*y_9_0+y_19_12*y_9_9^2 20 y_29_24*y_19_12+y_34_21*y_9_9+y_34_21+y_31_21*y_9_9+y_31_21+y_19_12*y_9_9+y_19_12+y_15_9*y_9_9^3+y_15_9*y_9_9^2*y_9_0+y_15_9*y_9_9^2+y_12_9*y_9_9^3*y_9_0 21 y_29_24*y_15_9 +y_26_24*y_9_9*y_9_0+y_23_15*y_9_9^2+y_23_15*y_9_9 22 y_29_24*y_12_9 +y_29_24*y_9_9+y_29_24*y_9_0+y_29_24+y_26_24*y_9_9+y_26_24*y_9_0+y_26_24+y_23_15*y_9_9^2+y_23_15*y_9_9*y_9_0+y_23_15*y_9_9+y_23_15*y_9_0+y_23_15 23 y_31_21^2 +y_29_24*y_9_0+y_26_24*y_9_9^2*y_9_0^2+y_26_24*y_9_9*y_9_0^3+y_23_15*y_9_9^3*y_9_0+y_23_15*y_9_9^2*y_9_0^2+y_23_15 24 y_31_21*y_29_24+y_34_21*y_9_9^2+y_34_21*y_9_9*y_9_0+y_34_21*y_9_0+y_34_21+y_19_12*y_9_9^3*y_9_0+y_19_12*y_9_9^2*y_9_0^2+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9*y_9_0^2+y_15_9*y_9_9^4*y_9_0+y_15_9*y_9_9^3*y_9_0^2+y_15_9*y_9_9^3+y_15_9*y_9_9^2*y_9_0+y_15_9*y_9_9^2+y_12_9*y_9_9^3*y_9_0+y_9_9^5*y_9_0+y_9_9^3*y_9_0^3+y_9_9^4*y_9_0+y_9_9^3*y_9_0^2 25 y_31_21*y_26_24 +y_34_21*y_9_9^2+y_34_21*y_9_9*y_9_0+y_34_21*y_9_9+y_34_21*y_9_0+y_31_21*y_9_9*y_9_0+y_31_21*y_9_0+y_15_9*y_9_9^3*y_9_0+y_15_9*y_9_9^2*y_9_0^2+y_15_9*y_9_9^3+y_12_9*y_9_9^4*y_9_0+y_12_9*y_9_9^3*y_9_0^2+y_12_9*y_9_9^2*y_9_0+y_9_9^5*y_9_0+y_9_9^4*y_9_0^2 26 y_31_21*y_23_15 +y_31_21*y_9_9*y_9_0+y_31_21*y_9_0+y_15_9*y_9_9^3*y_9_0+y_15_9*y_9_9^2*y_9_0^2+y_15_9*y_9_9^2*y_9_0+y_12_9*y_9_9^2*y_9_0+y_9_9^4*y_9_0^2+y_9_9^3*y_9_0^3 27 y_31_21*y_19_12 +y_29_24*y_9_9*y_9_0+y_29_24*y_9_0^2+y_29_24*y_9_0+y_26_24*y_9_9*y_9_0+y_26_24*y_9_0^2+y_23_15*y_9_9^2*y_9_0+y_23_15*y_9_9*y_9_0^2+y_23_15*y_9_9*y_9_0+y_23_15*y_9_0^2+y_23_15 28 y_31_21*y_15_9 +y_34_21+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9*y_9_0^2 29 y_31_21*y_12_9 +y_34_21*y_9_9+y_34_21*y_9_0+y_31_21 30 y_34_21^2 +y_29_24*y_9_9^2*y_9_0^2+y_29_24*y_9_9*y_9_0^3+y_29_24*y_9_9*y_9_0^2+y_26_24*y_9_9^2*y_9_0^2+y_26_24*y_9_9*y_9_0^3+y_23_15*y_9_9^3*y_9_0^2+y_23_15*y_9_9^2*y_9_0^3+y_23_15*y_9_9^2*y_9_0^2+y_23_15*y_9_9*y_9_0^3+y_23_15*y_9_9*y_9_0 31 y_34_21*y_31_21 +y_29_24*y_9_9^2*y_9_0^2+y_29_24*y_9_9*y_9_0^3+y_29_24*y_9_0+y_26_24*y_9_9*y_9_0^2+y_26_24*y_9_0+y_23_15*y_9_9*y_9_0^2+y_23_15*y_9_9*y_9_0+y_23_15*y_9_0 32 y_34_21*y_29_24 +y_31_21*y_9_9^2*y_9_0+y_31_21*y_9_9*y_9_0+y_19_12*y_9_9^3*y_9_0+y_19_12*y_9_9^2*y_9_0^2+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9*y_9_0^2+y_15_9*y_9_9^3*y_9_0+y_12_9*y_9_9^4*y_9_0+y_12_9*y_9_9^3*y_9_0^2+y_12_9*y_9_9^3*y_9_0+y_9_9^5*y_9_0^2+y_9_9^4*y_9_0^3 33 y_34_21*y_26_24 +y_34_21*y_9_9*y_9_0+y_34_21*y_9_0+y_19_12*y_9_9^3*y_9_0+y_19_12*y_9_9^2*y_9_0^2+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9*y_9_0^2+y_15_9*y_9_9^4*y_9_0+y_15_9*y_9_9^3*y_9_0^2+y_15_9*y_9_9^2*y_9_0+y_12_9*y_9_9^4*y_9_0+y_9_9^4*y_9_0^2+y_9_9^3*y_9_0^3 34 y_34_21*y_23_15 +y_34_21*y_9_9*y_9_0+y_34_21*y_9_0+y_15_9*y_9_9^2*y_9_0+y_12_9*y_9_9^3*y_9_0^2+y_9_9^4*y_9_0^2+y_9_9^3*y_9_0^3 35 y_34_21*y_19_12 +y_29_24*y_9_0+y_26_24*y_9_9*y_9_0^2+y_26_24*y_9_0+y_23_15*y_9_9^2*y_9_0+y_23_15*y_9_9*y_9_0 + y_23_15*y_9_0 36 y_34_21*y_15_9 +y_31_21*y_9_9*y_9_0 37 y_34_21*y_12_9 +y_34_21+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9*y_9_0^2 38 y_34_30^2 +y_29_24*y_9_9^4+y_29_24*y_9_9^3*y_9_0+y_29_24*y_9_9^2*y_9_0^2+y_29_24*y_9_9*y_9_0^3+y_29_24*y_9_9^3+y_29_24*y_9_9*y_9_0^2+y_26_24*y_9_9^4+y_26_24*y_9_9^3*y_9_0+y_26_24*y_9_9^2*y_9_0^2+y_26_24*y_9_9*y_9_0^3+y_26_24*y_9_9^2+y_26_24*y_9_9*y_9_0+y_23_15*y_9_9^5+y_23_15*y_9_9^4*y_9_0+y_23_15*y_9_9^3*y_9_0^2+y_23_15*y_9_9^2*y_9_0^3+y_23_15*y_9_9^4+y_23_15*y_9_9^3*y_9_0+y_23_15*y_9_9^2*y_9_0^2+y_23_15*y_9_9*y_9_0^3+y_23_15*y_9_9^2+y_23_15*y_9_9*y_9_0 39 y_34_30*y_34_21 +y_29_24*y_9_9^3*y_9_0+y_29_24*y_9_9*y_9_0^3+y_26_24*y_9_9^3*y_9_0+y_26_24*y_9_9*y_9_0^3+y_23_15*y_9_9^4*y_9_0+y_23_15*y_9_9^2*y_9_0^3+y_23_15*y_9_9^3*y_9_0+y_23_15*y_9_9*y_9_0^3 40 y_34_30*y_31_21 +y_29_24*y_9_9^3*y_9_0+y_29_24*y_9_9*y_9_0^3+y_23_15*y_9_9^3+y_23_15*y_9_9*y_9_0^2 41 y_34_30*y_29_24 +y_34_30*y_9_9+y_34_30+y_31_21*y_9_9^3+y_31_21*y_9_9^2*y_9_0+y_31_21*y_9_9^2+y_31_21*y_9_9*y_9_0+y_19_12*y_9_9^4+y_19_12*y_9_9^2*y_9_0^2+y_19_12*y_9_9^2*y_9_0+y_19_12*y_9_9*y_9_0^2+y_19_12*y_9_9^2+y_19_12*y_9_9*y_9_0+y_12_9*y_9_9^5+y_12_9*y_9_9^3*y_9_0^2+y_12_9*y_9_9^4+y_12_9*y_9_9^3*y_9_0+y_9_9^6*y_9_0+y_9_9^4*y_9_0^3+y_9_9^5+y_9_9^3*y_9_0^2+y_9_9^4+y_9_9^3*y_9_0 42 y_34_30*y_26_24 +y_34_30*y_9_9+y_34_30+y_34_21*y_9_9^2+y_34_21*y_9_9*y_9_0+y_34_21*y_9_9+y_34_21*y_9_0+y_31_21*y_9_9+y_31_21+y_19_12*y_9_9^4+y_19_12*y_9_9^2*y_9_0^2+y_19_12*y_9_9^3+y_19_12*y_9_9*y_9_0^2+y_19_12*y_9_9+y_19_12+y_15_9*y_9_9^5+y_15_9*y_9_9^3*y_9_0^2+y_12_9*y_9_9^5+y_12_9*y_9_9^4*y_9_0+y_12_9*y_9_9^4+y_12_9*y_9_9^3*y_9_0+y_9_9^5*y_9_0+y_9_9^3*y_9_0^3+y_9_9^5+y_9_9^4*y_9_0+y_9_9^4+y_9_9^3*y_9_0 43 y_34_30*y_23_15 +y_34_21*y_9_9^2+y_34_21*y_9_9*y_9_0+y_34_21*y_9_9+y_34_21*y_9_0+y_31_21*y_9_9+y_31_21+y_19_12*y_9_9+y_19_12+y_12_9*y_9_9^4*y_9_0+y_12_9*y_9_9^3*y_9_0^2+y_9_9^5*y_9_0+y_9_9^3*y_9_0^3+y_9_9^4*y_9_0+y_9_9^3*y_9_0^2 44 y_34_30*y_19_12 +y_26_24*y_9_9^2*y_9_0+y_26_24*y_9_9*y_9_0^2+y_23_15*y_9_9^3+y_23_15*y_9_9^2*y_9_0 45 y_34_30*y_15_9 +y_31_21*y_9_9^2+y_31_21*y_9_9*y_9_0+y_19_12*y_9_9^2+y_19_12*y_9_9*y_9_0 46 y_34_30*y_12_9 +y_19_12*y_9_9^3+y_19_12*y_9_9*y_9_0^2 /////delta=x_9_9^3*x_9_0^3+x_9_9^2*x_9_0^4+x_9_9^3*x_9_0^2+x_9_9*x_9_0^4+x_9_9^2*x_9_0^2+x_9_9*x_9_0^3 ///phi, the map from the integral closure to the ring*delta //1 x_19_12*x_15_9*x_9_9^4*x_9_0^2 + x_19_12*x_15_9*x_9_9^2*x_9_0^4 + // x_19_12*x_15_9*x_9_9^4*x_9_0 + x_19_12*x_15_9*x_9_9^3*x_9_0^2 + // x_19_12*x_15_9*x_9_9^2*x_9_0^3 + x_19_12*x_15_9*x_9_9*x_9_0^4 + // x_19_12*x_12_9*x_9_9^3*x_9_0^2 + x_19_12*x_12_9*x_9_9^2*x_9_0^3 + // x_19_12*x_15_9*x_9_9^3*x_9_0 + x_19_12*x_15_9*x_9_9*x_9_0^3 + // x_19_12*x_12_9*x_9_9^3*x_9_0 + x_19_12*x_12_9*x_9_9*x_9_0^3 + // x_19_12*x_9_9^3*x_9_0^2 + x_19_12*x_9_9^2*x_9_0^3 + // x_19_12*x_12_9*x_9_9^2*x_9_0 + x_19_12*x_12_9*x_9_9*x_9_0^2 + // x_19_12*x_9_9^3*x_9_0 + x_19_12*x_9_9*x_9_0^3 + x_19_12*x_9_9^2*x_9_0 + // x_19_12*x_9_9*x_9_0^2 //2 x_19_12*x_15_9*x_9_9^3*x_9_0^3 + x_19_12*x_15_9*x_9_9^2*x_9_0^4 + // x_19_12*x_15_9*x_9_9^3*x_9_0^2 + x_19_12*x_15_9*x_9_9*x_9_0^4 + // x_19_12*x_15_9*x_9_9^2*x_9_0^2 + x_19_12*x_15_9*x_9_9*x_9_0^3 //3 x_19_12*x_12_9*x_9_9^3*x_9_0^3 + x_19_12*x_12_9*x_9_9^2*x_9_0^4 + // x_19_12*x_12_9*x_9_9^3*x_9_0^2 + x_19_12*x_12_9*x_9_9*x_9_0^4 + // x_19_12*x_12_9*x_9_9^2*x_9_0^2 + x_19_12*x_12_9*x_9_9*x_9_0^3 //4 x_19_12^2*x_9_9^3*x_9_0^2 + x_19_12^2*x_9_9^2*x_9_0^3 + // x_19_12^2*x_9_9^3*x_9_0 + x_19_12^2*x_9_9*x_9_0^3 + // x_19_12^2*x_12_9*x_9_9*x_9_0 + x_19_12^2*x_15_9*x_9_9 + // x_19_12^2*x_15_9*x_9_0 + x_19_12^2*x_9_9*x_9_0 + x_19_12^2*x_12_9 + // x_19_12^2 //5 x_19_12^2*x_15_9*x_9_9^2*x_9_0 + x_19_12^2*x_15_9*x_9_9*x_9_0^2 + // x_19_12^2*x_12_9*x_9_9^2*x_9_0 + x_19_12^2*x_9_9^3*x_9_0 + // x_19_12^2*x_9_9^2*x_9_0^2 + x_19_12^2*x_15_9*x_9_9*x_9_0 + // x_19_12^2*x_15_9*x_9_0^2 + x_19_12^2*x_12_9*x_9_9*x_9_0 + // x_19_12^2*x_9_9^2*x_9_0 + x_19_12^2*x_15_9*x_9_9 + x_19_12^2*x_15_9*x_9_0 + // x_19_12^2*x_12_9*x_9_0 + x_19_12^2*x_9_9*x_9_0 + x_19_12^2*x_12_9 + // x_19_12^2*x_9_0 + x_19_12^2 //6 x_19_12^2*x_12_9*x_9_9*x_9_0^2 + x_19_12^2*x_9_9^2*x_9_0^2 + // x_19_12^2*x_9_9*x_9_0^3 + x_19_12^2*x_15_9*x_9_9*x_9_0 + // x_19_12^2*x_15_9*x_9_0^2 + x_19_12^2*x_9_9*x_9_0^2 + x_19_12^2*x_12_9*x_9_0 // + x_19_12^2*x_9_0 //7 x_19_12*x_9_9^3*x_9_0^3 + x_19_12*x_9_9^2*x_9_0^4 + x_19_12*x_9_9^3*x_9_0^2 + // x_19_12*x_9_9*x_9_0^4 + x_19_12*x_9_9^2*x_9_0^2 + x_19_12*x_9_9*x_9_0^3 //8 x_15_9*x_9_9^3*x_9_0^3 + x_15_9*x_9_9^2*x_9_0^4 + x_15_9*x_9_9^3*x_9_0^2 + // x_15_9*x_9_9*x_9_0^4 + x_15_9*x_9_9^2*x_9_0^2 + x_15_9*x_9_9*x_9_0^3 //9 x_12_9*x_9_9^3*x_9_0^3 + x_12_9*x_9_9^2*x_9_0^4 + x_12_9*x_9_9^3*x_9_0^2 + // x_12_9*x_9_9*x_9_0^4 + x_12_9*x_9_9^2*x_9_0^2 + x_12_9*x_9_9*x_9_0^3 //10 x_9_9^4*x_9_0^3 + x_9_9^3*x_9_0^4 + x_9_9^4*x_9_0^2 + x_9_9^2*x_9_0^4 + // x_9_9^3*x_9_0^2 + x_9_9^2*x_9_0^3 //11 x_9_9^3*x_9_0^4 + x_9_9^2*x_9_0^5 + x_9_9^3*x_9_0^3 + x_9_9*x_9_0^5 + // x_9_9^2*x_9_0^3 + x_9_9*x_9_0^4 ///psi, the map from the ring to its integral closure //psi= //1 f_19_12 //2 f_15_9 //3 f_12_9 //4 f_9_9 //5 f_9_0 //Total time: 1.040 seconds, Total memory usage: 20.69MB The new normal_p in SINGULAR thinks there are 4 components of the normalization: LIB "normal.lib"; > ring r=2,(z19,y15,y12,x9,u9),wp(19,15,12,9,9); > ideal i=y15^3+x9*u9*y15+x9^3*u9^2+x9^2*u9^3,y15^2+y12*x9*u9,z19^3+(y12+y15)*(x9+1)*u9*z19+(y12*(x9*u9+1)+y15*(x9+u9))*x9^2*u9; // ** redefining i ** > list nor=normal_p(i); // 'normal_p' computed a list, say nor, of generators of the normalization // of 4 component(s) (the associated primes P_i of the input ideal) // and its delta invariant(s): // nor[1] is a list of 4 list(s), where each list nor[1][i] consists of // elements g1..gk of the basering such that gj/gk generate the // normalization of the i-th component as (basering/P_i)-module in the // quotient field of basering/P_i. // nor[2] shows the delta-invariant of the input ideal and its components // (-1 means infinite). > nor[1]; [1]: [1]: y12^2+y15*x9+y15*u9+y12 [2]: y15*y12+x9^2*u9+x9*u9^2+y15 [3]: y15^2+y12*x9*u9 [4]: z19^3+y12*x9^3*u9^2+z19*y15*x9*u9+y15*x9^3*u9+y15*x9^2*u9^2+z19*y12*x9*u9+z19*y15*u9+z19*y12*u9+y12*x9^2*u9 [5]: z19*y15*x9^2*u9^3+z19*y15*x9*u9^4+z19*y15*x9^2*u9^2+z19*y15*x9*u9^3+z19*y12*u9^4+z19*y12*u9^3+z19*u9^4+z19*u9^3 [6]: z19*x9^3*u9^4+z19*x9^2*u9^5+z19*x9^3*u9^3+z19*x9^2*u9^4+z19*y15*x9*u9^3+z19*y15*u9^4+z19*y15*x9*u9^2+z19*y15*u9^3 [7]: x9^5*u9^5+x9^4*u9^6+y15*x9^4*u9^4+y15*x9^3*u9^5+x9^5*u9^4+x9^4*u9^5+y15*x9^4*u9^3+y15*x9^2*u9^5+y12*x9^3*u9^4+y12*x9^2*u9^5+y15*x9^3*u9^3+y15*x9^2*u9^4+y12*x9^3*u9^3+y12*x9^2*u9^4 [8]: y12*x9^5*u9^4+y12*x9^4*u9^5+x9^6*u9^4+x9^4*u9^6+y12*x9^5*u9^3+y12*x9^4*u9^4+x9^6*u9^3+x9^3*u9^6+y15*x9^4*u9^3+y15*x9^2*u9^5+x9^4*u9^4+x9^3*u9^5+y15*x9^4*u9^2+y15*x9^2*u9^4 [9]: y15*x9^5*u9^4+y15*x9^4*u9^5+x9^6*u9^4+x9^4*u9^6+y15*x9^5*u9^3+y15*x9^4*u9^4+y12*x9^4*u9^4+y12*x9^3*u9^5+x9^6*u9^3+x9^5*u9^4+y15*x9^4*u9^3+y15*x9^2*u9^5+y12*x9^4*u9^3+y12*x9^3*u9^4+x9^5*u9^3+x9^4*u9^4+y15*x9^4*u9^2+y15*x9^3*u9^3+y15*x9^3*u9^2+y15*x9^2*u9^3 [10]: z19^2*x9^4*u9^3+z19^2*x9^2*u9^5+z19^2*x9^4*u9^2+z19^2*x9^2*u9^4+z19^2*y15*x9*u9^3+z19^2*y15*u9^4+z19^2*y15*x9*u9^2+z19^2*y15*u9^3+z19^2*y12*u9^3+z19^2*y12*u9^2+z19^2*u9^3+z19^2*u9^2 [11]: z19*y12*x9*u9^3+z19*y12*x9*u9^2+z19*u9^4+z19*u9^3 [2]: [1]: 1 [3]: [1]: x9 [2]: y15 [3]: z19^2 [4]: u9 [4]: [1]: 1 //Macaulay2's integralClosure times out after a week or so and 5.1 GB's R=ZZ/2[z,y1,y2,x2,x1,MonomialOrder=>{Weights=>{19,15,12,9,9},Weights=>{12,9,9,9,0}}] I=ideal(y1^2+y2*x2*x1,y2*y1+y1+x2^2*x1+x2*x1^2,y2^2+y2+y1*(x2+x1),z^3+(y2+y1)*(x2+1)*x1*z+(y2*(x2*x1+1)+y1*(x2+x1))*x2^2*x1); S=R/I; integralClosure(S); bash: line 1: 6904 Virtual timer expired LPDEST="lab" PATH="$PATH:/usr/local/bin" "M2" "--no-readline" "--print-width" "79" /usr/local/sbin/appserv_: line 103: unexpected EOF while looking for matching `"' /usr/local/sbin/appserv_: line 107: syntax error: unexpected end of file Process M2 exited abnormally with code 2 //Its icFracP gives the following, which seems to be incomplete. i1 : R=ZZ/2[z,y1,y2,x2,x1,MonomialOrder=>{Weights=>{19,15,12,9,9},Weights=>{12, 9,9,9,0}}] o1 = R o1 : PolynomialRing i2 : I:=ideal(y1^2+y2*x2*x1,y2*y1+y1+x2^2*x1+x2*x1^2,y2^2+y2+y1*(x2+x1),z^3+(y2+ y1)*(x2+1)*x1*z+(y2*(x2*x1+1)+y1*(x2+x1))*x2^2*x1); o2 : Ideal of R i3 : S=R/I o3 = S o3 : QuotientRing i4 : icFracP S 2 2 2 2 2 2 z y1 + z y2 + y1 x2*x1 + z*y1*x1 + y1 y2 + z*y2*x1 + y1 x1 + z + y1*x2 o4 = {1, ----------------------------------------------------------------------- y1*x1 + y2*x1 --------------------------------------------------------------------------- 2 *x1 + y1*x1 + y2*x1 2 2 --------------------, y1 x2 + y1*x2 + y1*x2*x1 + y1*x1 + y2*x2 + y2*x1, --------------------------------------------------------------------------- 2 2 2 2 2 2 3 2 z x2 + z x2*x1 + z y1 + y1 y2*x2 + y1*y2*x2 + y1 + y1 y2 + y1*y2*x1 + y1 --------------------------------------------------------------------------- 2 y1 + y1*y2 --------------------------------------------------------------------------- 2 2 2 2 *y2 z*y1*x2 + z*y2*x2 + z x2 + z*y2*x2 + z*x2 + z*x2*x1 + y1*x2*x1 + z* ---, ---------------------------------------------------------------------- x2*x1 + y1 --------------------------------------------------------------------------- x2 + y1*y2 + x2*x1 + y1 -----------------------} o4 : List i5 :