//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 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+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