\documentclass{article} \usepackage{amssymb} \usepackage{amsmath} \usepackage{url} \urlstyle{same} \def\ul{\underline} \def\ov{\overline} \begin{document} \begin{center}{\Large Annotated Examples 8}\end{center} \begin{center}{\large Greuel, Laplagne, Seelich }\end{center} This seems to be on page 95 of Vasconcelas (unhomogenized), {\em Computational Methods in Commutative Algebra and Algebraic Geometry} with no attribution. It is example $I_7$ from Greuel, Laplagne, and Seelich, given by (homogenized) generators \[ x^2+zw,\ y^3+xwt,\ xw^3+z^3t+ywt^2,\ y^2w^4-xy^2z^2t-w^3t^3\] This is clearly homogeneous, maybe not immediately clear that it is a curve. Dehomogenize it by using $Z_{21}:=z/t$, $X_{18}:=x/t$, $W_{15}:=w/t$, $Y_{11}:=y/t$, and look at the divisors: \[div(Z_{21})=-21\cdot P_{\infty}+12\cdot Q+9\cdot R\] \[div(X_{18})=-18\cdot P_{\infty}+5\cdot Q+13\cdot R\] \[div(W_{15})=-15\cdot P_{\infty}-2\cdot Q+17\cdot R\] \[div(Y_{11})=-11\cdot P_{\infty}+1\cdot Q+10\cdot R\] Maybe it is no surprise that there exists $V_7$ with \[div(V_{7})=-7\cdot P_{\infty}+4\cdot Q+3\cdot R\] and $Z_{21}=V_7^3$, $X_{18}=Y_{11}V_7$. The one-point version would use $U_{37}:=W_{15}Y_{11}^2$ in place of $W_{15}$. and a Gr\"obner basis for an almost module presentation is: \[ U_{37}^2 - Y_{11}*f_7^9 - U_{37},\] \[ U_{37}f_{27} - Y_{11}^2*f_7^6 - f_{27},\] \[ Y_{11}^3*f_7^3 - U_{37}*f_{17} + f_{17},\] \[ f_{27}^2 - U_{37}*f_{17},\] \[ U_{37}*Y_{11} - f_{27}*f_7^3,\] \[ Y_{11}^4 - U_{37}*f_7 + f_7,\] \[ f_{27}*f_{17} - U_{37}*f_7,\] \[ f_{27}Y_{11} - f_{17}*f_7^3,\] \[ f_{17}^2 - f_{27}*f_7,\] \[ f_{17}Y_{11} - f_7^4.\] If one is then wedded to the original ``two-point'' presentation, replace $U_{37}$ by $W_{15}Y_{11}^2$ and then magically produce some more elements (see below). A straigh-forward attempt at this in SINGULAR 3-1-0 gives: \begin{verbatim} SINGULAR / A Computer Algebra System for Polynomial Computations / version 3-1-0 0< by: G.-M. Greuel, G. Pfister, H. Schoenemann \ Mar 2009 FB Mathematik der Universitaet, D-67653 Kaiserslautern \ > LIB "normal.lib"; > ring r=0,(u,z,x,w,y),wp(37,21,18,15,11); > ideal i=x2+zw,y3+xw,xw3+z3+yw,y2w4-xy2z2-w3; > list nor=normal(i); > nor; [1]: [1]: // characteristic : 0 // number of vars : 15 // block 1 : ordering dp // : names T(1) T(2) T(3) T(4) T(5) T(6) T(7) T(8) T(9) T(10) // block 2 : ordering wp // : names u z x w y // : weights 37 21 18 15 11 // block 3 : ordering C [2]: [1]: _[1]=w2 _[2]=xw _[3]=wy2 _[4]=xy2 _[5]=zwy _[6]=zxy _[7]=z2w _[8]=z2x _[9]=z3+wy _[10]=z2y2 _[11]=y3 > def R=nor[1][1]; > setring R; > normap; normap[1]=u normap[2]=z normap[3]=x normap[4]=w normap[5]=y > norid; norid[1]=T(2)+1 norid[2]=-T(3)*y+w norid[3]=T(8)*w+T(10)*y norid[4]=-T(4)*y+x norid[5]=T(1)*x+T(3)*y norid[6]=T(3)*x-T(4)*w norid[7]=T(4)*x+T(5)*y norid[8]=T(5)*x-T(6)*w norid[9]=T(6)*x+T(7)*y norid[10]=T(7)*x+T(10)*y norid[11]=-T(1)*y+T(8)*x+T(9)*w norid[12]=T(1)*z-T(4)*y norid[13]=T(3)*z-T(5)*y norid[14]=T(4)*z-T(6)*y norid[15]=T(5)*z-T(7)*y norid[16]=T(6)*z-T(8)*y norid[17]=T(1)*y+T(7)*z-T(9)*w norid[18]=T(8)*z-T(9)*x-y norid[19]=T(4)*w+y^2 norid[20]=T(1)*y^2-T(3)*w norid[21]=T(4)*y^2-T(5)*w norid[22]=T(6)*y^2-T(7)*w norid[23]=T(7)*y^2-T(10)*w norid[24]=T(8)*y^2-T(10)*x norid[25]=-T(3)*y+T(9)*y^2-T(10)*z norid[26]=T(3)*w*y-T(9) norid[27]=T(6)*w+z*y norid[28]=-T(1)*w*y+T(9)*w^2-T(10)*x*y norid[29]=T(3)*y^2+T(9)*x*w+T(10)*z*y norid[30]=-T(10)*y+z^2 norid[31]=T(5)*w*y^2-T(9)*z norid[32]=T(4)*w^2*y-T(9)*x norid[33]=-T(9)*z^2+T(10)*w^2*y norid[34]=-T(1)*w+T(3)*w^3-T(10)*x norid[35]=T(3)*y+T(4)*w^3+T(10)*z norid[36]=T(5)*w^3-T(9)*x*y norid[37]=T(6)*w^3+T(9)*z*y norid[38]=T(7)*w^3-T(9)*z*x norid[39]=-T(7)*w^2-T(10)*z^2*x+T(10)*w^4 norid[40]=T(1)^2-T(1)*w^4*y-T(10)*w*y^2-T(10) norid[41]=T(1)*T(2)+T(1) norid[42]=T(2)^2-1 norid[43]=T(1)*T(3)-T(1)*w^2*y+T(8)*y norid[44]=T(2)*T(3)+T(3) norid[45]=T(3)^2-T(1)*y norid[46]=T(1)*T(4)+T(3) norid[47]=T(2)*T(4)+T(4) norid[48]=T(3)*T(4)+y norid[49]=T(4)^2+T(5) norid[50]=T(1)*T(5)+y norid[51]=T(2)*T(5)+T(5) norid[52]=T(3)*T(5)-T(1)*z norid[53]=T(4)*T(5)+z norid[54]=T(5)^2-T(4)*z norid[55]=T(1)*T(6)+T(5) norid[56]=T(2)*T(6)+T(6) norid[57]=T(3)*T(6)+z norid[58]=T(4)*T(6)+T(7) norid[59]=T(5)*T(6)+T(10) norid[60]=T(6)^2+T(1)*w*y^2-T(1) norid[61]=T(1)*T(7)+z norid[62]=T(2)*T(7)+T(7) norid[63]=T(3)*T(7)-T(4)*z norid[64]=T(4)*T(7)+T(10) norid[65]=T(5)*T(7)-T(6)*z norid[66]=T(6)*T(7)-T(1)*x*w*y-T(3) norid[67]=T(7)^2-T(1)*z*w^2-y norid[68]=T(1)*T(8)+T(7) norid[69]=T(2)*T(8)+T(8) norid[70]=T(3)*T(8)+T(10) norid[71]=T(4)*T(8)+T(1)*w*y^2-T(1) norid[72]=T(5)*T(8)-T(1)*x*w*y-T(3) norid[73]=T(6)*T(8)+T(1)*z*w*y-T(4) norid[74]=T(7)*T(8)-T(1)*z*x*w-T(5) norid[75]=T(8)^2+T(1)*z^2*w-T(6) norid[76]=T(1)*T(9)-T(1)*w^2 norid[77]=T(2)*T(9)-T(1)*x*w norid[78]=T(3)*T(9)-T(1)*w*y^2 norid[79]=T(4)*T(9)-T(1)*x*y^2 norid[80]=T(5)*T(9)-T(1)*z*w*y norid[81]=T(6)*T(9)-T(1)*z*x*y norid[82]=T(7)*T(9)-T(1)*z^2*w norid[83]=T(8)*T(9)-T(1)*z^2*x norid[84]=T(9)^2-T(1)*z^3-T(1)*w*y norid[85]=T(1)*T(10)-T(4)*z norid[86]=T(2)*T(10)+T(10) norid[87]=T(3)*T(10)-T(5)*z norid[88]=T(4)*T(10)-T(6)*z norid[89]=T(5)*T(10)-T(7)*z norid[90]=T(6)*T(10)-T(1)*z*w^2-y norid[91]=T(7)*T(10)-T(1)*z*w*y^2+T(4)*y norid[92]=T(8)*T(10)-T(1)*z*x*y^2-z norid[93]=T(9)*T(10)-T(1)*z^2*y^2 norid[94]=T(10)^2+T(1)*z*x*w*y+T(3)*z norid[95]=x^2+z*w norid[96]=x*w+y^3 norid[97]=z^3+x*w^3+w*y norid[98]=-z^3*x+z*w^4-x*w*y norid[99]=-z^2*x*y^2+w^4*y^2-w^3 > option(redSB); > ideal j=std(norid);j; j[1]=x*w+y^3 j[2]=x^2+z*w j[3]=z*w^2-x*y^3 j[4]=z^3-w^2*y^3+w*y j[5]=z^2*x*y^2-w^4*y^2+w^3 j[6]=w^5*y^2+z^2*y^5-w^4 j[7]=T(10)*y-z^2 j[8]=T(10)*z-w^2*y^2+w j[9]=T(10)*w^2-z*x*y^2 j[10]=T(9)-w^2 j[11]=T(8)*w+z^2 j[12]=T(8)*z+w*y^3-y j[13]=T(8)*y^2-T(10)*x j[14]=T(7)*w-z*x j[15]=T(7)*x+z^2 j[16]=T(7)*z+T(8)*x j[17]=T(7)*y^2-T(10)*w j[18]=T(6)*w+z*y j[19]=T(6)*x+T(7)*y j[20]=T(6)*z-T(8)*y j[21]=T(6)*y^2-z*x j[22]=T(5)*w-x*y j[23]=T(5)*x+z*y j[24]=T(5)*z-T(7)*y j[25]=T(5)*y^2-z*w j[26]=T(4)*y-x j[27]=T(4)*w+y^2 j[28]=T(4)*x+T(5)*y j[29]=T(4)*z-T(6)*y j[30]=T(3)*y-w j[31]=T(3)*w-T(8)*x*y-w^3*y j[32]=T(3)*x+y^2 j[33]=T(3)*z-T(5)*y j[34]=T(2)+1 j[35]=T(1)*y-T(8)*x-w^3 j[36]=T(1)*w+T(10)*x-w^5*y-z^2*y^4 j[37]=T(1)*x+w j[38]=T(1)*z-x j[39]=T(10)^2+T(5)*y-x*y^4 j[40]=T(8)*T(10)+z*w*y^2-z j[41]=T(7)*T(10)+y^5+x j[42]=T(6)*T(10)+w*y^3-y j[43]=T(5)*T(10)+T(8)*x j[44]=T(4)*T(10)-T(8)*y j[45]=T(3)*T(10)-T(7)*y j[46]=T(1)*T(10)-T(6)*y j[47]=T(8)^2-T(6)-z*y^3 j[48]=T(7)*T(8)-T(5)+x*y^3 j[49]=T(6)*T(8)-T(4)-y^4 j[50]=T(5)*T(8)-T(3)+w^2*y j[51]=T(4)*T(8)-T(1)-z^2*x*y+w^4*y j[52]=T(3)*T(8)+T(10) j[53]=T(1)*T(8)+T(7) j[54]=T(7)^2+w*y^3-y j[55]=T(6)*T(7)-T(3)+w^2*y j[56]=T(5)*T(7)-T(8)*y j[57]=T(4)*T(7)+T(10) j[58]=T(3)*T(7)-T(6)*y j[59]=T(1)*T(7)+z j[60]=T(6)^2-T(1)-z^2*x*y+w^4*y j[61]=T(5)*T(6)+T(10) j[62]=T(4)*T(6)+T(7) j[63]=T(3)*T(6)+z j[64]=T(1)*T(6)+T(5) j[65]=T(5)^2-T(6)*y j[66]=T(4)*T(5)+z j[67]=T(3)*T(5)-x j[68]=T(1)*T(5)+y j[69]=T(4)^2+T(5) j[70]=T(3)*T(4)+y j[71]=T(1)*T(4)+T(3) j[72]=T(3)^2-T(8)*x-w^3 j[73]=T(1)*T(3)+T(8)*y-w^5-z^2*y^3 j[74]=T(1)^2-T(10)-w^7-z*x*y^6-z^2*w*y \end{verbatim} Now try the same with $V_7$: \begin{verbatim} > ring r=0,(w,y,u),wp(15,11,7); > ideal i=y2+uw,yuw3+u9+yw,y2w4-y3u7-w3; > list nor=normal(i); > nor; [1]: [1]: // characteristic : 0 // number of vars : 3 // block 1 : ordering wp // : names w y u // : weights 15 11 7 // block 2 : ordering C [2]: // characteristic : 0 // number of vars : 8 // block 1 : ordering dp // : names T(1) T(2) T(3) T(4) T(5) // block 2 : ordering wp // : names w y u // : weights 15 11 7 // block 3 : ordering C [2]: [1]: _[1]=1 [2]: _[1]=-w3y-u8 _[2]=-w4+yu7 _[3]=wu _[4]=yu4 _[5]=u7 _[6]=-wu > def R=nor[1][1]; > setring R; > normap; normap[1]=w normap[2]=y normap[3]=u > norid; norid[1]=u norid[2]=y norid[3]=w > setring r; > def R2=nor[1][2]; > setring R2; > normap; normap[1]=w normap[2]=y normap[3]=u > norid; norid[1]=T(3)+1 norid[2]=T(1)*y+T(2)*u norid[3]=T(1)*w-T(2)*y norid[4]=T(1)*u^2+y norid[5]=T(2)*u^2+w norid[6]=T(4)*u^3-T(5)*y norid[7]=T(4)*y*u^2+T(5)*w norid[8]=T(1)+T(2)*w*y*u+T(5)*u norid[9]=T(2)*w^2*u+T(2)-T(5)*y norid[10]=-T(1)*u-T(5)*u^2+w^2*y norid[11]=-T(2)*u+T(5)*y*u+w^3 norid[12]=-T(4)*y+u^4 norid[13]=T(4)*w+y*u^3 norid[14]=T(4)*w^3+T(5)*u^5-y*u^2 norid[15]=T(1)^2+T(2)*w^2-T(4)*u^2 norid[16]=T(1)*T(2)+T(1)*w^5+T(2)*w*u^8+u^5 norid[17]=T(2)^2-T(1)*w^2*u^7+T(1)*u^6+T(2)*w^5 norid[18]=T(1)*T(3)+T(1) norid[19]=T(2)*T(3)+T(2) norid[20]=T(3)^2-1 norid[21]=T(1)*T(4)+u^2 norid[22]=T(2)*T(4)+T(1)*u^3 norid[23]=T(3)*T(4)+T(4) norid[24]=T(4)^2+T(1)*w^2*u+T(1) norid[25]=T(1)*T(5)+T(4)*u norid[26]=T(2)*T(5)-u^4 norid[27]=T(3)*T(5)+T(5) norid[28]=T(4)*T(5)+T(2)*w*u^4-u norid[29]=T(5)^2-T(1)*w*u^6-T(4) norid[30]=y^2+w*u norid[31]=w^3*y*u+u^9+w*y norid[32]=-w^4*u+y*u^8-w^2 > option(redSB); > ideal j=std(norid);j; j[1]=y^2+w*u j[2]=w^3*y*u+u^9+w*y j[3]=w^4*u-y*u^8+w^2 j[4]=T(5)*w+u^6 j[5]=T(5)*u^3-w^2*y*u-y j[6]=T(5)*y*u^2+w^3*u+w j[7]=T(4)*y-u^4 j[8]=T(4)*w+y*u^3 j[9]=T(4)*u^3-T(5)*y j[10]=T(3)+1 j[11]=T(2)-T(5)*y+w^5-w*y*u^7 j[12]=T(1)+T(5)*u+w^4*y+w*u^8 j[13]=T(5)^2-T(4)+w*y*u^4 j[14]=T(4)*T(5)-w^2*u^2-u j[15]=T(4)^2-T(5)*u \end{verbatim} Ignoring the silly first ring, we see that there are three unnecessary variables $T(1)=-T(5)u-w^4y-wu^8$, $T(2)=T(5)y-w^5+wyu^7$, and $T(3)=-1$. So \begin{verbatim} > ring r1=0,(f27,f17,f15,f11,f7),wp(27,17,15,11,7); > map phi=R2,-f27*f7-f15^4*f11-f15*f7^8,f27*f11-f15^5+f15*f11*f7^7,-1,f17,f27,f15,f11,f7; > ideal i1=phi(j); > option(redSB); > ideal j1=std(i1);j1; j1[1]=f11^2+f15*f7 j1[2]=f17*f11-f7^4 j1[3]=f17*f15+f11*f7^3 j1[4]=f17^2-f27*f7 j1[5]=f27*f11-f17*f7^3 j1[6]=f27*f15+f7^6 j1[7]=f27*f17-f15^2*f7^2-f7 j1[8]=f15^2*f11*f7-f27*f7^3+f11 j1[9]=f15^3*f7+f17*f7^5+f15 j1[10]=f27^2+f15*f11*f7^4-f17 \end{verbatim} or \begin{verbatim} > intmat A[5][5]=1,1,1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0,0,0,27,17,15,11,7; > ring r1=0,(f27,f17,f15,f11,f7),M(A); > map phi=R2,-f27*f7-f15^4*f11-f15*f7^8,f27*f11-f15^5+f15*f11*f7^7,-1,f17,f27,f15,f11,f7; > ideal i1=phi(j); > option(redSB); > ideal j1=std(i1);j1; j1[1]=f11^2+f15*f7 j1[2]=f17*f11-f7^4 j1[3]=f27*f11-f17*f7^3 j1[4]=f17*f15+f11*f7^3 j1[5]=f27*f15+f7^6 j1[6]=f17^2-f27*f7 j1[7]=f27*f17-f15^2*f7^2-f7 j1[8]=f27^2+f15*f11*f7^4-f17 j1[9]=f15^2*f11*f7-f27*f7^3+f11 j1[10]=f15^3*f7+f17*f7^5+f15 \end[verbatim} take your pick. Let's try Macaulay2: \begin{verbatim} Macaulay 2, version 1.2 with packages: Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, SchurRings, TangentCone i1 : R=QQ[z,x,w,y] o1 = R o1 : PolynomialRing i2 : I=ideal (x^2+z*w,y^3+x*w,x*w^3+z^3+y*w,y^2*w^4-x*y^2*z^2-w^3) 2 3 3 3 4 2 2 2 3 o2 = ideal (x + z*w, y + x*w, x*w + z + w*y, w y - z x*y - w ) o2 : Ideal of R i3 : S=R/I o3 = S o3 : QuotientRing i4 : F=icFractions(S) 3 2 2 w - 99w -w 2 4 2 2 2 -w -y -y z*x 3 5 y - w + z x -z*y -z*y o4 = {--, ---, ---, ---, ---------, ---, --, ----------, -----, -----, x x w w x y w w x w 3 --------------------------------------------------------------------------- 5 2 5 2 - w + w y - z x*w*y - w + 11w y - 11z x*w*y 2 2 0 0 -z*x*y -11w -w*y --------------------, ------------------------, ------, -----, -----, x y w 11x x --------------------------------------------------------------------------- 2 z*x*y ------, z, x, w, y} w o4 : List i5 : P=presentation(integralClosure(S)) o5 = | 10w_9^11w_11^3w_14^4+10w_9^3w_11^2w_14^3+10w_9^5w_14^2 -w_9^2w_11^2w_14^2-w_9^3w_14^3 -w_9^3w_11^2w_14^2-w_9^4w_14^3 w_9^3w_11^2w_14^2+w_9^4w_14^3 w_9^4w_11^2w_14^2+w_9^5w_14^3 -w_9^9w_11^3w_14^3+w_9^2w_14^3-w_9^3w_14 w_9^2w_11^2w_14^2+w_9^3w_14^3 -w_9^8w_11^3w_14^3+w_9w_14^3-w_9^2w_14 -w_9^2w_11w_13w_14+99w_9^2w_14^3-100w_9^3w_14 -w_9w_11w_13w_14+99w_9w_14^3-100w_9^2w_14 w_9^2w_11w_13w_14^2-99w_9^2w_14^4+100w_9^3w_14^2 -w_9w_11^2w_14^2-w_9^2w_14^3 w_9^2w_13w_14^2+99w_9w_11w_14^3-100w_9^2w_11w_14 w_9^3w_11^2w_14+w_9^4w_14^2 -w_9^2w_11^2w_14-w_9^3w_14^2 -w_9w_11^2w_14-w_9^2w_14^2 w_9^9w_11^3w_14^3+w_9w_11^2w_14^2+w_9^3w_14 w_9^3w_11^2w_14+w_9^4w_14^2 w_9^2w_11^2w_14+w_9^3w_14^2 100w_9^10w_11^3w_14^3-w_9^3w_11w_13w_14+w_9^2w_11^2w_14^2 -99w_9^8w_11^2w_14^3+w_9w_13w_14-w_9w_11 w_9^8w_11w_14^3+w_14^2-w_9 -99w_9^7w_11^2w_14^3+w_13w_14-w_11 w_9^8w_11^2w_14^3+w_11w_14^2-w_9w_11 10000w_9^27w_11^8w_14^9-10000w_9^11w_11^6w_14^7-20000w_9^13w_11^4w_14^6- 199w_9^6w_11^3w_14^3+w_13^2+w_14 -99w_9^7w_11^3w_14^3+w_11w_13w_14+w_9w_14 99w_9^8w_11w_14^3+w_11w_13+w_9 w_11^2w_14+w_9w_14^2 -w_9w_11^2w_14-w_9^2w_14^2 w_11^2+w_9w_14 -w_9w_11^2-w_9^2w_14 w_9^2w_11^2+w_9^3w_14 1 32 o5 : Matrix (QQ[w , w , w , w ]) <--- (QQ[w , w , w , w ]) 9 11 13 14 9 11 13 14 i6 : g=gens gb ideal(P) o6 = | w_11w_13-99w_14^2+100w_9 w_11^2+w_9w_14 w_9w_13w_14+99w_11w_14^2-100w_9w_11 9801w_9^7w_11w_14^4-10000w_9^8w_11w_14^2-w_13^2-w_14 99w_9^8w_14^4+w_13w_14-w_11 w_9^8w_11w_14^3+w_14^2-w_9 100w_9^9w_14^3+w_9w_13-w_11w_14 10000w_9^9w_11w_14^2+w_9w_13^2+9801w_14^3-9800w_9w_14 1000000w_9^10w_14^2-w_9w_13^3-9801w_13w_14^3-1960200w_11w_14^2+1970000w_9w_11 100000000w_9^10w_11w_14-w_9w_13^4-9801w_13^2w_14^3-194059800w_14^4+490050000w_9w_14^2-296000000w_9^2 10000000000w_9^11w_14+w_9w_13^5+9801w_13^3w_14^3+194059800w_13w_14^4+48514950000w_11w_14^3+395000000w_9^2w_13-49104000000w_9w_11w_14 1000000000000w_9^11w_11+w_9w_13^6+9801w_13^4w_14^3+194059800w_13^2w_14^4+ 4802980050000w_14^5+494000000w_9^2w_13^2-8742492000000w_9w_14^3+ 3940200000000w_9^2w_14 100000000000000w_9^12-w_9w_13^7-9801w_13^5w_14^3- 194059800w_13^3w_14^4-4802980050000w_13w_14^5-593000000w_9^2w_13^3- 769447107000000w_11w_14^4+973239300000000w_9w_11w_14^2-198990000000000w_9^2w_11 970299w_9^7w_14^6+1000000w_9^9w_14^2-w_13^3+19899w_13w_14-19900w_11 96059601w_9^6w_11w_14^7-100000000w_9^9w_11w_14+w_13^4-29799w_13^2w_14+ 1960200w_14^2-1990000w_9 9509900499w_9^6w_14^9+10000000000w_9^10w_14+w_13^5-39699w_13^3w_14+ 198960300w_13w_14^2-1990000w_9w_13-197010000w_11w_14 941480149401w_9^5w_11w_14^10-1000000000000w_9^10w_11-w_13^6+49599w_13^4w_14 -493970400w_13^2w_14^2+1990000w_9w_13^2+38909970000w_14^3-39402000000w_9w_14 93206534790699w_9^5w_14^12+100000000000000w_9^11-w_13^7+59499w_13^5w_14- 886990500w_13^3w_14^2+1990000w_9w_13^3+2008616940000w_13w_14^3+ 3900798000000w_11w_14^2-5910300000000w_9w_11 9227446944279201w_9^4w_11w_14^13-100000000000000w_9^11w_13+w_13^8-69399w_13^6w_14+1378020600w_13^4w_14^2-100990000w_9w_13^4-7869222900000w_13^2w_14^3- 19405980000000000w_14^4+49490149500000000w_9w_14^2-30092040000000000w_9^2 913517247483640899w_9^4w_14^15-100000000000000w_9^11w_13^2+w_13^9-79299w_13^7w_14+1967060700w_13^5w_14^2-199990000w_9w_13^5-17620727850000w_13^3w_14^3 -18925681995000000w_13w_14^4-9702504850500000000w_11w_14^3- 69394050000000000w_9^2w_13+9790806960000000000w_9w_11w_14 90438207500880449001w_9^3w_11w_14^16+100000000000000w_9^11w_13^3-w_13^10+ 89199w_13^8w_14-2654110800w_13^6w_14^2+298990000w_9w_13^6+32233430790000w_13^4w_14^3-29968654914000000w_13^2w_14^4+1437944985249300000000w_14^5+ 118497060000000000w_9^2w_13^2-2803116187080000000000w_9w_14^3+ 1365259698000000000000w_9^2w_14 8953382542587164451099w_9^3w_14^18+100000000000000w_9^11w_13^4-w_13^11+ 99099w_13^9w_14-3439170900w_13^7w_14^2+397990000w_9w_13^7+52677630720000w_13^5w_14^3-165796930728000000w_13^3w_14^4+2110294950528600000000w_13w_14^5+177401070000000000w_9^2w_13^3+353492607507930000000000w_11w_14^4-511243751007000000000000w_9w_11w_14^2+155640860100000000000000w_9^2w_11 886384871716129280658801w_9^2w_11w_14^19-100000000000000w_9^11w_13^5+w_13^12-108999w_13^10w_14+4322241000w_13^8w_14^2-496990000w_9w_13^8-79923626640000w_13^6w_14^3+436685005548000000w_13^4w_14^4-3456896861187000000000w_13^2w_14^5-246106080000000000w_9^2w_13^4-41557884784610040000000000w_14^6+99238213197090000000000000w_9w_14^4-72238760550000000000000000w_9^2w_14^2+14554975050000000000000000w_9^3 --------------------------------------------------------------------------- 87752102299896798785221299w_9^2w_14^21-100000000000000w_9^11w_13^6+w_13^13- --------------------------------------------------------------------------- 118899w_13^11w_14+5303321100w_13^9w_14^2-595990000w_9w_13^9- --------------------------------------------------------------------------- 114941717550000w_13^7w_14^3+900364699575000000w_13^5w_14^4- --------------------------------------------------------------------------- 6338396712384000000000w_13^3w_14^5-324612090000000000w_9^2w_13^5- --------------------------------------------------------------------------- 45229657367273940000000000w_13w_14^6-10762297335315306000000000000w_11w_14^ --------------------------------------------------------------------------- 5+18016938763560000000000000000w_9w_11w_14^3+9654965100000000000000000w_9^ --------------------------------------------------------------------------- 3w_13-7219073074950000000000000000w_9^2w_11w_14 --------------------------------------------------------------------------- 8687458127689783079736908601w_9w_11w_14^22+100000000000000w_9^11w_13^7-w_13 --------------------------------------------------------------------------- ^14+128799w_13^12w_14-6382411200w_13^10w_14^2+694990000w_9w_13^10+ --------------------------------------------------------------------------- 158702202450000w_13^8w_14^3-1624173793110000000w_13^6w_14^4+ --------------------------------------------------------------------------- 12091886514279000000000w_13^4w_14^5+412919100000000000w_9^2w_13^6+ --------------------------------------------------------------------------- 39335991632023680000000000w_13^2w_14^6+1121204944001053296000000000000w_14^ --------------------------------------------------------------------------- 7-2915077407878869200000000000000w_9w_14^5+106242840000000000000000w_9^3w_ --------------------------------------------------------------------------- 13^2+2467501222211190000000000000000w_9^2w_14^3- --------------------------------------------------------------------------- 673589328192000000000000000000w_9^3w_14 --------------------------------------------------------------------------- 860058354641288524893953951499w_9w_14^24+100000000000000w_9^11w_13^8-w_13^ --------------------------------------------------------------------------- 15+138699w_13^13w_14-7559511300w_13^11w_14^2+793990000w_9w_13^11+ --------------------------------------------------------------------------- 212175380340000w_13^9w_14^3-2685056026554000000w_13^7w_14^4+ --------------------------------------------------------------------------- 22626003297021300000000w_13^5w_14^5+511027110000000000w_9^2w_13^7+ --------------------------------------------------------------------------- 8681397967527120000000000w_13^3w_14^6+1159140003660867789000000000000w_13w_ --------------------------------------------------------------------------- 14^7+281005651448045152200000000000000w_11w_14^6+ --------------------------------------------------------------------------- 15698947770000000000000000w_9^3w_13^3-510116198112629460000000000000000w_9w --------------------------------------------------------------------------- _11w_14^4+287625595757841000000000000000000w_9^2w_11w_14^2- --------------------------------------------------------------------------- 59674164739200000000000000000000w_9^3w_11 --------------------------------------------------------------------------- 85145777109487563964501441198401w_11w_14^25-100000000000000w_9^11w_13^9+w_ --------------------------------------------------------------------------- 13^16-148599w_13^14w_14+8834621400w_13^12w_14^2-892990000w_9w_13^12- --------------------------------------------------------------------------- 276331550220000w_13^10w_14^3+4169561100408000000w_13^8w_14^4- --------------------------------------------------------------------------- 40516028115740100000000w_13^6w_14^5-618936120000000000w_9^2w_13^8+ --------------------------------------------------------------------------- 75166824138125940000000000w_13^4w_14^6-1264649801043939057000000000000w_13^ --------------------------------------------------------------------------- 2w_14^7-25774757250504199126200000000000000w_14^8- --------------------------------------------------------------------------- 38093448690000000000000000w_9^3w_13^4+71481089499945428160000000000000000w_ --------------------------------------------------------------------------- 9w_14^6-70507657606451868000000000000000000w_9^2w_14^4+ --------------------------------------------------------------------------- 30002842720055700000000000000000000w_9^3w_14^2- --------------------------------------------------------------------------- 5202782049960000000000000000000000w_9^4 --------------------------------------------------------------------------- 8429431933839268832485642678641699w_14^27-100000000000000w_9^11w_13^10+w_13 --------------------------------------------------------------------------- ^17-158499w_13^15w_14+10207741500w_13^13w_14^2-991990000w_9w_13^13- --------------------------------------------------------------------------- 352141011090000w_13^11w_14^3+6173844675273000000w_13^9w_14^4- --------------------------------------------------------------------------- 69098985055534500000000w_13^7w_14^5-736646130000000000w_9^2w_13^9+ --------------------------------------------------------------------------- 259536881795323770000000000w_13^5w_14^6-1673546869170735570000000000000w_13 --------------------------------------------------------------------------- ^3w_14^7-22243413304928548903500000000000000w_13w_14^8- --------------------------------------------------------------------------- 68260044600000000000000000w_9^3w_13^5- --------------------------------------------------------------------------- 6002645310649941183450000000000000000w_11w_14^7+ --------------------------------------------------------------------------- 11810134119719012220000000000000000000w_9w_11w_14^5- --------------------------------------------------------------------------- 8238354527039907600000000000000000000w_9^2w_11w_14^3- --------------------------------------------------------------------------- 4827189010050000000000000000000000w_9^4w_13+ --------------------------------------------------------------------------- 2457934646547600000000000000000000000w_9^3w_11w_14 | 1 28 o6 : Matrix (QQ[w , w , w , w ]) <--- (QQ[w , w , w , w ]) 9 11 13 14 9 11 13 14 \end{verbatim} I haven't a clue as to what $w_0$, $w_3$, and $w_5$ are in icFractions; nor do I have a clue as to what $w_9$, $w_{11}$, $w_{13}$, and $w_{14}$ are in the presentation, let alone whether the two answers are related or correct in any sense. \newpage Now consider the fact that there are magical elements $A:=Y/V$, $B:=A/V$, $C:=B/V$ with divisors \[div(A)=-4\cdot P_{\infty}-3\cdot Q+7\cdot R\] \[div(B)=3\cdot P_{\infty}-7\cdot Q+4\cdot R\] \[div(C)=10\cdot P_{\infty}-11\cdot Q+1\cdot R\] which told me they clearly existed and had poles only at $P_{\infty}$ and $Q$. So everything in the integral closure can be written in terms of $V$ and $C$ with \[ C^7V^{11}+C^3-V=0,\] namely, $Y=CV^3$, $W=-C^2V^5$, $A=CV^2$, $B=CV$, (and $Z=V^3$, $X=YV=CV^4$). Let's analyze the answer in SINGULAR 3-1-0: \begin{verbatim} SINGULAR / A Computer Algebra System for Polynomial Computations / version 3-1-0 0< by: G.-M. Greuel, G. Pfister, H. Schoenemann \ Mar 2009 FB Mathematik der Universitaet, D-67653 Kaiserslautern \ > LIB "/home/leonada/presolve.lib"; > LIB "/home/leonada/normal.lib"; > ring r=0,(w,y,u),wp(15,11,7); > ideal i=y7+y3u-u11,y2+wu; > list nor=normal(i); > nor; [1]: [1]: // characteristic : 0 // number of vars : 8 // block 1 : ordering dp // : names T(1) T(2) T(3) T(4) T(5) // block 2 : ordering wp // : names w y u // : weights 15 11 7 // block 3 : ordering C [2]: // characteristic : 0 // number of vars : 3 // block 1 : ordering wp // : names w y u // : weights 15 11 7 // block 2 : ordering C [2]: [1]: _[1]=-w3y-u8 _[2]=-w4+yu7 _[3]=wu _[4]=yu4 _[5]=u7 _[6]=-wu [2]: _[1]=1 > def R=nor[1][1]; > setring R; > normap; normap[1]=w normap[2]=y normap[3]=u > norid; norid[1]=T(3)+1 norid[2]=T(1)*y+T(2)*u norid[3]=T(1)*w-T(2)*y norid[4]=T(1)*u^2+y norid[5]=T(2)*u^2+w norid[6]=T(4)*u^3-T(5)*y norid[7]=T(4)*y*u^2+T(5)*w norid[8]=T(1)+T(2)*w*y*u+T(5)*u norid[9]=T(2)*w^2*u+T(2)-T(5)*y norid[10]=-T(1)*u-T(5)*u^2+w^2*y norid[11]=-T(2)*u+T(5)*y*u+w^3 norid[12]=-T(4)*y+u^4 norid[13]=T(4)*w+y*u^3 norid[14]=T(4)*w^3+T(5)*u^5-y*u^2 norid[15]=T(1)^2+T(2)*w^2-T(4)*u^2 norid[16]=T(1)*T(2)+T(1)*w^5+T(2)*w*u^8+u^5 norid[17]=T(2)^2-T(1)*w^2*u^7+T(1)*u^6+T(2)*w^5 norid[18]=T(1)*T(3)+T(1) norid[19]=T(2)*T(3)+T(2) norid[20]=T(3)^2-1 norid[21]=T(1)*T(4)+u^2 norid[22]=T(2)*T(4)+T(1)*u^3 norid[23]=T(3)*T(4)+T(4) norid[24]=T(4)^2+T(1)*w^2*u+T(1) norid[25]=T(1)*T(5)+T(4)*u norid[26]=T(2)*T(5)-u^4 norid[27]=T(3)*T(5)+T(5) norid[28]=T(4)*T(5)+T(2)*w*u^4-u norid[29]=T(5)^2-T(1)*w*u^6-T(4) norid[30]=y^2+w*u norid[31]=w^3*y*u+u^9+w*y norid[32]=-w^4*u+y*u^8-w^2 > option(redSB); > ideal j=std(norid);j; j[1]=y^2+w*u j[2]=w^3*y*u+u^9+w*y j[3]=w^4*u-y*u^8+w^2 j[4]=T(5)*w+u^6 j[5]=T(5)*u^3-w^2*y*u-y j[6]=T(5)*y*u^2+w^3*u+w j[7]=T(4)*y-u^4 j[8]=T(4)*w+y*u^3 j[9]=T(4)*u^3-T(5)*y j[10]=T(3)+1 j[11]=T(2)-T(5)*y+w^5-w*y*u^7 j[12]=T(1)+T(5)*u+w^4*y+w*u^8 j[13]=T(5)^2-T(4)+w*y*u^4 j[14]=T(4)*T(5)-w^2*u^2-u j[15]=T(4)^2-T(5)*u \end{verbatim} It is not too hard to figure out that $T(3)=-1$ $T(2)=T(5)y-w^5+wyu^7$, and $T(1)=-T(5)u-w^4y-wu^8$ may not be necessary. $T(4)=-yu^3/w=V/C$, $T(5)=-u^6/w=V/C^2$. So can $C$ be recovered from this answer? I'm guessing not; at least the following is the best I can do. \begin{verbatim} > ring r1=0,(c,T(4),T(5),w,y,u),lp; > map phi=R,-T(5)*u-w^4*y-w*u^8,T(5)*y-w^5+w*y*u^7,-1,T(4),T(5),w,y,u; > ideal i1=phi(j); > ideal j1=i1,c^7*u^11+c^3-u; > option(redSB); > ideal k1=std(j1);k1; k1[1]=y^7+y^3*u-u^11 k1[2]=w*u+y^2 k1[3]=w*y^5-y^3+u^10 k1[4]=w^2*y^3-w*y-u^9 k1[5]=w^3*y^2-w^2+y*u^8 k1[6]=T(5)*u^3+w*y^3-y k1[7]=T(5)*y*u^2-w^2*y^2+w k1[8]=T(5)*y^2-u^7 k1[9]=T(5)*w+u^6 k1[10]=T(5)^2*y-y^4*u^3-u^4 k1[11]=T(5)^3-y^4-y*u^10-u k1[12]=T(4)-T(5)^2+y^3*u^3 k1[13]=c^7*u^11+c^3-u k1[14]=c^7*y*u^8+c^3*T(5)-c^3*w^6*y+c^3*y^6*u^5-c^3*y^2*u^6-T(5)*u -w^4*y+y^2*u^7 k1[15]=c^7*y^2*u^5+c^3*T(5)^2+c^3*w^11-3*c^3*y^4*u^12-5*c^3*y^3*u^3 +c^3*y*u^22+3*c^3*u^13-T(5)^2*u+w^9+y^4*u^13+4*y^3*u^4-2*u^14 k1[16]=c^7*y^3-c^7*u^10-c^3*w^21*y-9*c^3*w^2-35*c^3*y^6*u^16 -28*c^3*y^5*u^7+6*c^3*y^4*u^35+35*c^3*y^3*u^26+21*c^3*y^2*u^17 -c^3*y*u^45+8*c^3*y*u^8-15*c^3*u^36-w^19*y-8*w*y^2+20*y^6*u^17 +21*y^5*u^8-y^4*u^36-15*y^3*u^27-15*y^2*u^18-7*y*u^9+5*u^37 k1[17]=c^7*w*y+c^7*u^9-c^3*w^23*y-10*c^3*w^4+56*c^3*y^6*u^15 -c^3*y^5*u^43+36*c^3*y^5*u^6-21*c^3*y^4*u^34-70*c^3*y^3*u^25 -28*c^3*y^2*u^16+6*c^3*y*u^44+c^3*y*u^7+35*c^3*u^35-w^21*y-9*w^2 -35*y^6*u^16-28*y^5*u^7+6*y^4*u^35+35*y^3*u^26+21*y^2*u^17 -y*u^45+8*y*u^8-15*u^36 k1[18]=c^7*w^2+c^3*T(5)-c^3*w^24-11*c^3*w^6*y+c^3*y^6*u^42 -25*c^3*y^6*u^5+21*c^3*y^5*u^33+70*c^3*y^4*u^24+84*c^3*y^3*u^15 -6*c^3*y^2*u^43-2*c^3*y^2*u^6-35*c^3*y*u^34-56*c^3*u^25-T(5)*u -w^22-10*w^4*y+28*y^6*u^6-6*y^5*u^34-35*y^4*u^25-56*y^3*u^16 +y^2*u^44+2*y^2*u^7+15*y*u^35+35*u^26 \end{verbatim} \newpage Versions of icFractions and integralClosure in Macaulay2 version 1.2 don't do well; but the icFractions in Macaulay2 version 1.2.1 is supposed to produce $a:=x/y=V$,$b:=-z^2/w=V/C^2$, and $c:=-w/x=CV$. At least here I think I can recover $V=a$ and $C=b(1-c^7a^3)$, though I need to recheck this. \newpage With MAGMA's Normalisation (and Ideal ?), it is possible to reverse the order of the variables and weight them to get a fairly quick reasonable answer to the type I subproblem: : \begin{verbatim} t:=Cputime(); P := PolynomialRing(RationalField(),2); I := Ideal(y^7+y^3*x-x^11); Js:=Normalisation(I); N := Js[1][1]; N<[a]> := N; N; G:=GroebnerBasis(N);G; Cputime(t); tt:=Cputime(); R:=PolynomialRing(Rationals(),5,"grevlexw",[37,27,17,11,7]); hNR:=homR|R.5,R.4,R.3,R.2,R.1>; GR:=[G[i]@hNR: i in [1..#G]]; IR:=ideal; BR:=GroebnerBasis(IR);BR; Cputime(tt); Loading file "11_7a" Ideal of Polynomial ring of rank 5 over Rational Field Order: Lexicographical Variables: a[1], a[2], a[3], a[4], a[5] Basis: [ -a[1]*a[5] + a[2]^4, -a[1]^3*a[4] + a[2]*a[5] + a[2], -a[1]^3*a[3] + a[2]*a[4], -a[1]^4 + a[2]*a[3], -a[1]^4*a[4] + a[1]*a[2] + a[2]^5, -a[1]^6*a[3] + a[2]^2*a[5] + a[2]^2, -a[1]^7 + a[2]^2*a[4], -a[1]^4*a[2] + a[2]^2*a[3], -a[1]^7*a[3] + a[1]*a[2]^2 + a[2]^6, -a[1]^10 + a[2]^3*a[5] + a[2]^3, -a[1]^7*a[2] + a[2]^3*a[4], -a[1]^4*a[2]^2 + a[2]^3*a[3], -a[1]^9*a[2] + a[5]^2 + a[5], -a[1]^6*a[2]^2 + a[4]*a[5], -a[1]^3*a[2]^3 + a[3]*a[5], -a[1]^3*a[2]^3 - a[3] + a[4]^2, -a[1]*a[5] - a[1] + a[3]*a[4], -a[1]*a[4] + a[3]^2 ] [ a[1] + a[2]^4 - a[3]*a[4], a[2]^15 - a[3] - a[4]^2*a[5]^3 + a[4]^2*a[5]^2 - a[4]^2*a[5] + a[4]^2, a[2]^11*a[3] - a[4]*a[5]^3, a[2]^7*a[3]^2 - a[5]^3 - a[5]^2, a[2]^4*a[4] + a[3]^2 - a[3]*a[4]^2, a[2]^4*a[5] + a[2]^4 + a[3]*a[4] - a[4]^3, a[2]^3*a[3]^4 - a[4]*a[5]^2 - a[4]*a[5], a[2]^3*a[3]^3*a[4] - a[5]^3 - 2*a[5]^2 - a[5], a[2]^3*a[3]^2*a[4]^3 - a[5]^4 - 3*a[5]^3 - 3*a[5]^2 - a[5], a[2]^2*a[3]^9 - a[4]*a[5]^4 - 3*a[4]*a[5]^3 - 3*a[4]*a[5]^2 - a[4]*a[5], a[2]^2*a[3]^8*a[4] - a[5]^5 - 4*a[5]^4 - 6*a[5]^3 - 4*a[5]^2 - a[5], a[2]^2*a[3]^7*a[4]^3 - a[5]^6 - 5*a[5]^5 - 10*a[5]^4 - 10*a[5]^3 - 5*a[5]^2 - a[5], a[2]*a[3]^14 - a[4]*a[5]^6 - 5*a[4]*a[5]^5 - 10*a[4]*a[5]^4 - 10*a[4]*a[5]^3 - 5*a[4]*a[5]^2 - a[4]*a[5], a[2]*a[3]^13*a[4] - a[5]^7 - 6*a[5]^6 - 15*a[5]^5 - 20*a[5]^4 - 15*a[5]^3 - 6*a[5]^2 - a[5], a[2]*a[3]^12*a[4]^3 - a[5]^8 - 7*a[5]^7 - 21*a[5]^6 - 35*a[5]^5 - 35*a[5]^4 - 21*a[5]^3 - 7*a[5]^2 - a[5], a[2]*a[4]^4 - a[3]^7, a[2]*a[4]^2*a[5] + a[2]*a[4]^2 - a[3]^6, a[2]*a[5]^2 + 2*a[2]*a[5] + a[2] - a[3]^5, a[3]^19 - a[4]*a[5]^8 - 7*a[4]*a[5]^7 - 21*a[4]*a[5]^6 - 35*a[4]*a[5]^5 - 35*a[4]*a[5]^4 - 21*a[4]*a[5]^3 - 7*a[4]*a[5]^2 - a[4]*a[5], a[3]^18*a[4] - a[5]^9 - 8*a[5]^8 - 28*a[5]^7 - 56*a[5]^6 - 70*a[5]^5 - 56*a[5]^4 - 28*a[5]^3 - 8*a[5]^2 - a[5], a[3]^17*a[4]^3 - a[5]^10 - 9*a[5]^9 - 36*a[5]^8 - 84*a[5]^7 - 126*a[5]^6 - 126*a[5]^5 - 84*a[5]^4 - 36*a[5]^3 - 9*a[5]^2 - a[5], a[3]^16*a[4]^5 - a[5]^11 - 10*a[5]^10 - 45*a[5]^9 - 120*a[5]^8 - 210*a[5]^7 - 252*a[5]^6 - 210*a[5]^5 - 120*a[5]^4 - 45*a[5]^3 - 10*a[5]^2 - a[5], a[3]^15*a[4]^7 - a[5]^12 - 11*a[5]^11 - 55*a[5]^10 - 165*a[5]^9 - 330*a[5]^8 - 462*a[5]^7 - 462*a[5]^6 - 330*a[5]^5 - 165*a[5]^4 - 55*a[5]^3 - 11*a[5]^2 - a[5], a[3]^14*a[4]^9 - a[5]^13 - 12*a[5]^12 - 66*a[5]^11 - 220*a[5]^10 - 495*a[5]^9 - 792*a[5]^8 - 924*a[5]^7 - 792*a[5]^6 - 495*a[5]^5 - 220*a[5]^4 - 66*a[5]^3 - 12*a[5]^2 - a[5], a[3]^13*a[4]^11 - a[5]^14 - 13*a[5]^13 - 78*a[5]^12 - 286*a[5]^11 - 715*a[5]^10 - 1287*a[5]^9 - 1716*a[5]^8 - 1716*a[5]^7 - 1287*a[5]^6 - 715*a[5]^5 - 286*a[5]^4 - 78*a[5]^3 - 13*a[5]^2 - a[5], a[3]^12*a[4]^13 - a[5]^15 - 14*a[5]^14 - 91*a[5]^13 - 364*a[5]^12 - 1001*a[5]^11 - 2002*a[5]^10 - 3003*a[5]^9 - 3432*a[5]^8 - 3003*a[5]^7 - 2002*a[5]^6 - 1001*a[5]^5 - 364*a[5]^4 - 91*a[5]^3 - 14*a[5]^2 - a[5], a[3]^11*a[4]^15 - a[5]^16 - 15*a[5]^15 - 105*a[5]^14 - 455*a[5]^13 - 1365*a[5]^12 - 3003*a[5]^11 - 5005*a[5]^10 - 6435*a[5]^9 - 6435*a[5]^8 - 5005*a[5]^7 - 3003*a[5]^6 - 1365*a[5]^5 - 455*a[5]^4 - 105*a[5]^3 - 15*a[5]^2 - a[5], a[3]^10*a[4]^17 - a[5]^17 - 16*a[5]^16 - 120*a[5]^15 - 560*a[5]^14 - 1820*a[5]^13 - 4368*a[5]^12 - 8008*a[5]^11 - 11440*a[5]^10 - 12870*a[5]^9 - 11440*a[5]^8 - 8008*a[5]^7 - 4368*a[5]^6 - 1820*a[5]^5 - 560*a[5]^4 - 120*a[5]^3 - 16*a[5]^2 - a[5], a[3]^9*a[4]^19 - a[5]^18 - 17*a[5]^17 - 136*a[5]^16 - 680*a[5]^15 - 2380*a[5]^14 - 6188*a[5]^13 - 12376*a[5]^12 - 19448*a[5]^11 - 24310*a[5]^10 - 24310*a[5]^9 - 19448*a[5]^8 - 12376*a[5]^7 - 6188*a[5]^6 - 2380*a[5]^5 - 680*a[5]^4 - 136*a[5]^3 - 17*a[5]^2 - a[5], a[3]^8*a[4]^21 - a[5]^19 - 18*a[5]^18 - 153*a[5]^17 - 816*a[5]^16 - 3060*a[5]^15 - 8568*a[5]^14 - 18564*a[5]^13 - 31824*a[5]^12 - 43758*a[5]^11 - 48620*a[5]^10 - 43758*a[5]^9 - 31824*a[5]^8 - 18564*a[5]^7 - 8568*a[5]^6 - 3060*a[5]^5 - 816*a[5]^4 - 153*a[5]^3 - 18*a[5]^2 - a[5], a[3]^7*a[4]^23 - a[5]^20 - 19*a[5]^19 - 171*a[5]^18 - 969*a[5]^17 - 3876*a[5]^16 - 11628*a[5]^15 - 27132*a[5]^14 - 50388*a[5]^13 - 75582*a[5]^12 - 92378*a[5]^11 - 92378*a[5]^10 - 75582*a[5]^9 - 50388*a[5]^8 - 27132*a[5]^7 - 11628*a[5]^6 - 3876*a[5]^5 - 969*a[5]^4 - 171*a[5]^3 - 19*a[5]^2 - a[5], a[3]^6*a[4]^25 - a[5]^21 - 20*a[5]^20 - 190*a[5]^19 - 1140*a[5]^18 - 4845*a[5]^17 - 15504*a[5]^16 - 38760*a[5]^15 - 77520*a[5]^14 - 125970*a[5]^13 - 167960*a[5]^12 - 184756*a[5]^11 - 167960*a[5]^10 - 125970*a[5]^9 - 77520*a[5]^8 - 38760*a[5]^7 - 15504*a[5]^6 - 4845*a[5]^5 - 1140*a[5]^4 - 190*a[5]^3 - 20*a[5]^2 - a[5], a[3]^5*a[4]^27 - a[5]^22 - 21*a[5]^21 - 210*a[5]^20 - 1330*a[5]^19 - 5985*a[5]^18 - 20349*a[5]^17 - 54264*a[5]^16 - 116280*a[5]^15 - 203490*a[5]^14 - 293930*a[5]^13 - 352716*a[5]^12 - 352716*a[5]^11 - 293930*a[5]^10 - 203490*a[5]^9 - 116280*a[5]^8 - 54264*a[5]^7 - 20349*a[5]^6 - 5985*a[5]^5 - 1330*a[5]^4 - 210*a[5]^3 - 21*a[5]^2 - a[5], a[3]^4*a[4]^29 - a[5]^23 - 22*a[5]^22 - 231*a[5]^21 - 1540*a[5]^20 - 7315*a[5]^19 - 26334*a[5]^18 - 74613*a[5]^17 - 170544*a[5]^16 - 319770*a[5]^15 - 497420*a[5]^14 - 646646*a[5]^13 - 705432*a[5]^12 - 646646*a[5]^11 - 497420*a[5]^10 - 319770*a[5]^9 - 170544*a[5]^8 - 74613*a[5]^7 - 26334*a[5]^6 - 7315*a[5]^5 - 1540*a[5]^4 - 231*a[5]^3 - 22*a[5]^2 - a[5], a[3]^3*a[4]^31 - a[5]^24 - 23*a[5]^23 - 253*a[5]^22 - 1771*a[5]^21 - 8855*a[5]^20 - 33649*a[5]^19 - 100947*a[5]^18 - 245157*a[5]^17 - 490314*a[5]^16 - 817190*a[5]^15 - 1144066*a[5]^14 - 1352078*a[5]^13 - 1352078*a[5]^12 - 1144066*a[5]^11 - 817190*a[5]^10 - 490314*a[5]^9 - 245157*a[5]^8 - 100947*a[5]^7 - 33649*a[5]^6 - 8855*a[5]^5 - 1771*a[5]^4 - 253*a[5]^3 - 23*a[5]^2 - a[5], a[3]^2*a[4]^33 - a[5]^25 - 24*a[5]^24 - 276*a[5]^23 - 2024*a[5]^22 - 10626*a[5]^21 - 42504*a[5]^20 - 134596*a[5]^19 - 346104*a[5]^18 - 735471*a[5]^17 - 1307504*a[5]^16 - 1961256*a[5]^15 - 2496144*a[5]^14 - 2704156*a[5]^13 - 2496144*a[5]^12 - 1961256*a[5]^11 - 1307504*a[5]^10 - 735471*a[5]^9 - 346104*a[5]^8 - 134596*a[5]^7 - 42504*a[5]^6 - 10626*a[5]^5 - 2024*a[5]^4 - 276*a[5]^3 - 24*a[5]^2 - a[5], a[3]*a[4]^35 - a[5]^26 - 25*a[5]^25 - 300*a[5]^24 - 2300*a[5]^23 - 12650*a[5]^22 - 53130*a[5]^21 - 177100*a[5]^20 - 480700*a[5]^19 - 1081575*a[5]^18 - 2042975*a[5]^17 - 3268760*a[5]^16 - 4457400*a[5]^15 - 5200300*a[5]^14 - 5200300*a[5]^13 - 4457400*a[5]^12 - 3268760*a[5]^11 - 2042975*a[5]^10 - 1081575*a[5]^9 - 480700*a[5]^8 - 177100*a[5]^7 - 53130*a[5]^6 - 12650*a[5]^5 - 2300*a[5]^4 - 300*a[5]^3 - 25*a[5]^2 - a[5], a[3]*a[5] + a[3] - a[4]^2, a[4]^37 - a[5]^27 - 26*a[5]^26 - 325*a[5]^25 - 2600*a[5]^24 - 14950*a[5]^23 - 65780*a[5]^22 - 230230*a[5]^21 - 657800*a[5]^20 - 1562275*a[5]^19 - 3124550*a[5]^18 - 5311735*a[5]^17 - 7726160*a[5]^16 - 9657700*a[5]^15 - 10400600*a[5]^14 - 9657700*a[5]^13 - 7726160*a[5]^12 - 5311735*a[5]^11 - 3124550*a[5]^10 - 1562275*a[5]^9 - 657800*a[5]^8 - 230230*a[5]^7 - 65780*a[5]^6 - 14950*a[5]^5 - 2600*a[5]^4 - 325*a[5]^3 - 26*a[5]^2 - a[5] ] 0.630 [ f37^2 - f11*f7^9 + f37, f37*f27 - f11^2*f7^6, f27^2 - f11^3*f7^3 - f17, f37*f17 - f11^3*f7^3, f37*f11 - f27*f7^3 + f11, f27*f17 - f37*f7 - f7, f11^4 - f37*f7, f27*f11 - f17*f7^3, f17^2 - f27*f7, f17*f11 - f7^4 ] 0.050 \end{verbatim} I did get integralClosure in Macaulay2 version 1.2 to finally give me: \begin{verbatim} i9 : toString(P#0) o9 = QQ[w_6, w_7, w_1, u]/(w_1^7*u^3+w_1^3+u^7,w_7*w_1^2*u^3-829440*w_1^8*u^2+ 4148928*w_1^6*u^19+1016064*w_1^5*u^9-282475249*w_1^2*u^53-69177612*w_1*u^43 -248832*w_1*u^6-16941456*u^33,w_6*u^2+282475249*w_1^6*u^50+69177612*w_1^5*u ^40-580608*w_1^5*u^3+16941456*w_1^4*u^30+4148928*w_1^3*u^20+282475249*w_1^2 *u^47+1016064*w_1^2*u^10+69177612*w_1*u^37+248832*w_1+16941456*u^27,w_6*w_1 -w_7*u^2,w_6*w_7-687970713600*w_1^11+79792266297612001*w_1^6*u^98- 2343938939766144*w_1^6*u^61-16862305517568*w_1^6*u^24+39081926349850776*w_1 ^5*u^88-615027710142720*w_1^5*u^51-7914959732736*w_1^5*u^14+ 14356626006067632*w_1^4*u^78-80330149896192*w_1^4*u^41-2036393312256*w_1^4* u^4+4687877879532288*w_1^3*u^68-2459086221312*w_1^3*u^31+79792266297612001* w_1^2*u^95-908874282766464*w_1^2*u^58-13248954335232*w_1^2*u^21+ 39081926349850776*w_1*u^85+134720355555072*w_1*u^48-1179869773824*w_1*u^11+ 14356626006067632*u^75+86068017745920*u^38+61917364224*u,w_6^2+282475249*w_ 6*w_1^6*u^48-16941456*w_7*w_1^6*u^23+69177612*w_7*w_1^4*u^40-580608*w_7*w_1 ^4*u^3+16941456*w_7*w_1^3*u^30-12792528*w_7*w_1^2*u^20+282475249*w_7*w_1*u^ 47+1016064*w_7*w_1*u^10+69177612*w_7*u^37+248832*w_7) \end{verbatim} Since I have no idea how to extract the ideal from this, given that the presentation command didn't work for some reason, I used MAGMA \begin{verbatim} Q:=Rationals(); P:=PolynomialRing(Q,4); e1:=w_1^7*u^3+w_1^3+u^7; e2:=w_7*w_1^2*u^3-829440*w_1^8*u^2+4148928*w_1^6*u^19+1016064*w_1^5*u^9-282475249*w_1^2*u^53-69177612*w_1*u^43-248832*w_1*u^6-16941456*u^33; e3:=w_6*u^2+282475249*w_1^6*u^50+69177612*w_1^5*u^40-580608*w_1^5*u^3+16941456*w_1^4*u^30+4148928*w_1^3*u^20+282475249*w_1^2*u^47+1016064*w_1^2*u^10+69177612*w_1*u^37+248832*w_1+16941456*u^27; e4:=w_6*w_1-w_7*u^2; e5:=w_6*w_7-687970713600*w_1^11+79792266297612001*w_1^6*u^98-2343938939766144*w_1^6*u^61-16862305517568*w_1^6*u^24+39081926349850776*w_1^5*u^88-615027710142720*w_1^5*u^51-7914959732736*w_1^5*u^14+14356626006067632*w_1^4*u^78-80330149896192*w_1^4*u^41-2036393312256*w_1^4*u^4+4687877879532288*w_1^3*u^68-2459086221312*w_1^3*u^31+79792266297612001*w_1^2*u^95-908874282766464*w_1^2*u^58-13248954335232*w_1^2*u^21+39081926349850776*w_1*u^85+134720355555072*w_1*u^48-1179869773824*w_1*u^11+14356626006067632*u^75+86068017745920*u^38+61917364224*u; e6:=w_6^2+282475249*w_6*w_1^6*u^48-16941456*w_7*w_1^6*u^23+69177612*w_7*w_1^4*u^40-580608*w_7*w_1^4*u^3+16941456*w_7*w_1^3*u^30-12792528*w_7*w_1^2*u^20+282475249*w_7*w_1*u^47+1016064*w_7*w_1*u^10+69177612*w_7*u^37+248832*w_7; I:=ideal; N:=function(g) l:=NormalForm(g,I); return l; end function; //reducuing w_7 NormalForm(N(e2*(w_1^5*u^3+w_1))+w_7*u^10,ideal) div 82944; //7*w_1^6*u^9 - 3*w_1^2*u^6/u^10 //reducing w_6 NormalForm(N(-(7*w_1^6*u^3-3*w_1^2)*(w_1^6*u^3+w_1^2)),ideal) div -3; //w_1*u^7/u^9 \end{verbatim} to see that (with $a:=w_1$) this can be reduced to $b:=(7*a^6*u^3-3*a^2)/u^4$ and $c:=a/u^2$, with $a^7*u^3+a^3+u^7=0$. And from there, $(b+3*c^2)/7=a^6/u=c^6*u^11$ and $a=c*u^2$; so $b$ and $a$ are not necessary, as advertised. And SINGULAR on \begin{verbatim} > ring r=13,(w,y,u),lp; > ideal i=y7+y3u-u11,wu+y2; > list nor=normalP(i,"withRing"); > nor; [1]: [1]: // characteristic : 13 // number of vars : 2 // block 1 : ordering dp // : names T(3) // block 2 : ordering lp // : names u // block 3 : ordering C [2]: // characteristic : 13 // number of vars : 1 // block 1 : ordering lp // : names w // block 2 : ordering C [2]: [1]: _[1]=yu5 _[2]=y2 _[3]=yu3 _[4]=u6 [2]: _[1]=1 [3]: [1]: 8,0 [2]: 10 > def R=nor[1][1]; > setring R; > normap; normap[1]=-T(3)^2*u^5 normap[2]=T(3)*u^3 normap[3]=u > norid; norid[1]=T(3)^15*u^33+T(3)^8*u^23+3*T(3)^4*u^12+T(3)^3-2*T(3)*u^13-u norid[2]=T(3)^7*u^15+T(3)^3*u^4-u^5 norid[3]=T(3)^7*u^14+T(3)^3*u^3-u^4 norid[4]=T(3)^8*u^16+T(3)^4*u^5-T(3)*u^6 norid[5]=2*T(3)^8*u^25+T(3)^7*u^13+2*T(3)^4*u^14+T(3)^3*u^2-2*T(3)*u^15-u^3 norid[6]=T(3)^24*u^55+3*T(3)^6*u^24+4*T(3)^5*u^12+T(3)^4-T(3)^3*u^25-3*T(3)^2*u^13-T(3)*u norid[7]=T(3)^7*u^21+T(3)^3*u^10-u^11 norid[8]=T(3)^7*u^20+T(3)^3*u^9-u^10 norid[9]=T(3)^7*u^19+T(3)^3*u^8-u^9 norid[10]=T(3)^8*u^21+T(3)^4*u^10-T(3)*u^11 > option(redSB); > ideal j=std(norid);j; j[1]=T(3)^7*u^11+T(3)^3-u \end{verbatim} works fairly well, though it slows down considerably, even by $q=23$. \end{document} i13 : g0=toString(f0#0) o13 = (-143503601609868434285603076356671071740077383739246066639249*u^372- 12646476735808344539036733807052891837594219369552679741568*u^335- 456789396865036058191147423864203005444805247224620101632*u^298- 11828504625087036067247009619918675748006864746115497984*u^261- 270107850824716256749265723967854868207481537551138816*u^224- 4940029116715707412716756694085066529216884455243776*u^187- 69462370331677059863183084774872386705181491855360*u^150- 673181831400007171266765457334237928540642213888*u^113- 3282785513178335674233503126220068468167802880*u^76- 3990681284838968584114021700626728251031552*u^39+ 7292200359928928450425352952773345280000*u^2)/(w_5) i14 : g1=toString(f0#1) o14 = (-143503601609868434285603076356671071740077383739246066639249*w_1*u^370- 12646476735808344539036733807052891837594219369552679741568*w_1*u^333- 456789396865036058191147423864203005444805247224620101632*w_1*u^296- 11828504625087036067247009619918675748006864746115497984*w_1*u^259- 270107850824716256749265723967854868207481537551138816*w_1*u^222- 4940029116715707412716756694085066529216884455243776*w_1*u^185- 69462370331677059863183084774872386705181491855360*w_1*u^148- 673181831400007171266765457334237928540642213888*w_1*u^111- 3282785513178335674233503126220068468167802880*w_1*u^74- 3990681284838968584114021700626728251031552*w_1*u^37+ 7292200359928928450425352952773345280000*w_1)/(w_5) i15 : g2=toString(f0#2) o15 = (7013917513256831509822245487375292336175042657159804835549682059258995650 85056460548921105038170450472159367478974893200773108684108799275271888242 90211209853092410507685419426170384516735561570286057073419746322113682392 2353968761276160898741176387037978403216031744000000*w_1^6*u^202+ 25803736690087849493225061093142878645619836779568875333230935987386857350 64350813391285223965776291004270044445886823667949741260341567704719433135 62874452414148698561902628473114317277209247147009790962662800384299484501 35465426749593661102942609801700523796909260800000*w_1^6*u^165+ 45497725429454899096830084112007858275843043393186209071741648839240115846 29789692304298539819769740054717618265346516096158768272943506106825365908 31425408138246950924239725522679810450698615407955748704888749456909465432 807672863209013567556308339934347435169546240000*w_1^6*u^128+ 60588287426844227504371584588311551622718995348642602913872447884099335191 79542046916133249053148572802973090337296121642471710384406384860719615386 65514778330536078258924530742320418285302180754469146563357792720099469170 5536729333870380063577132495550820143595520000*w_1^6*u^91+ 40205829274794016893243187517185111592845563471113373465285040667048161101 02048654826269195321996898756004794487156531344229667526184229793238332618 99802469751290609288319718423563453254592981405142815104817858967749754595 78793731727254288064780132099764791541760000*w_1^6*u^54+ 41382390662737995177137593316393945826217813272541314521701796184753582737 24594250174257339096226195390161934619039785989985529577520281931326875862 33241685875056527980690990979850480503094408130020763200399176327482044497 19810078714872385580435678896848896000000*w_1^6*u^17- 30344950584052279298617052904343023328293388961370869760126514123098048318 34901961801402904532877019570407477933926673463819376024357332055456070214 31164993948810241573602928716938083776165958051591928411921017861340295082 554049136450112043106014911279477047492281012060160000*w_1^5*u^266- 17827971205617502444733152825337994988921614675942460350873740160889910664 47045224351358437056944466954301055468325371824828050139003126177639619318 59362377765571861193613498057057521531067761056413858951899130078248359068 97751681961964610340493969916988205188880877486080000*w_1^5*u^229- 56912930678998289965414791954702372099248917560953844951317420137987278995 47315279882674109452583083831236010400824847686273224751054256976491893170 97713817093664130976647403343782548650082842456035434538605941585151022839 852934652006941986407125982625102475752437186560000*w_1^5*u^192- 12257859511018271867069506272787171880034031048313549489666228963812422528 10269814500892059420756344897132043358303692319290700398099611904446217666 27100360654336135596815367256041703664838316375756525100330563791363146647 99356039458688014856113468212775436223701319680000*w_1^5*u^155- 15444616293862365154913758559806449761967683714863503269647768053284486729 93399246589883020078070702303899611432303627289967292226085887338583268515 34932060583248154509939674137043628467693515209261231436121075892856771458 590228476631741729945165489440450320185425920000*w_1^5*u^118- 89856718369145698251593238273705233510289280533678172999633976753693833159 43388418818204255783054251132484043834969199994600614282201627369200482828 54983804756681705008987007569334299210837255563466513357648718843898094738 992674468407498530422408084993596823961600000*w_1^5*u^81- 12813592393004718370462023710038950344079268678100181568544737822842982546 74291432139220307895580613698162814029784843100216206896234963385103540522 09222383559658327396361705143572488098153348879330383748619613537489044753 3026235477227728526091186631143472496640000*w_1^5*u^44- 59614488349682586089513939951587917156616297631512097750350726796763864687 50916086685604810222654783275143243148676762530591399151289721869858644699 63853569087716486886949806933758195202536962492226789724464599751474733969 5530863680413789039198417127997440000000*w_1^5*u^7+ 18047725711797857075562328841066987635486447079066056650337635221900880049 39708455415411640013722775487939141395105076928918812562591532767676496570 61101045963723875454854219994913612508273631027768960338285386716365656550 556927183194818824471215865600621917458965966356480000*w_1^4*u^256+ 93557866385164735861573396751044871661979318999114952278637703468226936431 62336454321997628870262508798081340650103992083645744169695705888487798173 37761749207082653494181622456861629026011574056982350184893393996638619020 8732661087911457321542025029323256367342734540800000*w_1^4*u^219+ 18780121994679516205891400162115068459309501971823803967839148697444494477 37947094367723448591999246978516510474316448733498609374593356426850443621 81136953729096413440986265892113356093789911796357769143523892193088063874 720083064243739407989474167304862788779476254720000*w_1^4*u^182+ 23345107403126435062832667159070529862760019625568846001069739328907888596 78751347722966639867659913479610775788537663828609143185060876509685275736 90328527825689699782284510788718130758572606879735005119548425178274512200 5341699583401697362072948250217717022704271360000*w_1^4*u^145+ 22553980091600009664858895643476808681588432351928882828517121008259406012 89186013963267102678920879260960004021793316440055313322004746841893604965 25503282222966304022514763062476866981969170065463140065818535575202735540 68145214468054797300038100319771839074140160000*w_1^4*u^108+ 14249606297480031766626813348096990827986751497450401928168264872694972797 27477807215174125823971127888435038522985603556605252162124929310305114453 86892840966347563810964643326255978765431865105402510776581299681383817508 019223405140706814987079018352787293470720000*w_1^4*u^71+ 66384370020141644700402601681630817000481264806357291632322142781903845050 75325854817834881712522035490414072051679564077277411989227610206819510130 51896949805207561591449912751147751588650025485068810144325967415137369736 441348928855616267488030939859795312640000*w_1^4*u^34- 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91454842002838873055920927623592126130315895767040000*u^253+ 12728961412947583110418149217829234239725077414845571738590163737173732847 83991354329523486921124150856881815054435917290291937982271524610678611996 37790714177834374625058724143790697826668241368296918392502502584576682859 98275729371308105199376904801800348799105761280000*u^216+ 23797324464565809485101667559474544545608712588914185217015830888747841256 26837603756867798756270111722759312853041251257266957196562156831861971509 68346762102858429034940998236326531009836893138810740676782746133885323063 89190643566973165401211901168830654170875822080000*u^179+ 10232573542830390518094477990673087470632431484342644912824552185863819715 50383389540140861736169247910896443697887730845874027064263546203287405470 39194099180672353465797654235571556989720268598724859170795657280411799175 8780855383088834820896607103265364647419576320000*u^142+ 20147793277957852596932771198810110674120706243685100668339411629052966662 37245647171144530182368599280317207545196140140886803113819992519550145827 90034969087602899414930996982470583609472548896429690381447921782179664985 51819631441157881375231035624928735542640640000*u^105+ 17544741192495857356827985063862059958973873777546862206021693298005704236 89595320153790968230770212862606306408966607946841054471163724884703540608 50095761233673975913758488766853420909571966493174353726821721198229201695 300016781043710505175917102803982567342080000*u^68+ 70805135187373916486660274655875151132365128777902265809002617383890554071 32735910432957659293842052253105010780449560626950241154426164736520116865 56232625049588140758857773504462146485629489518282094395480375012739008585 540926923606279756745777981098976870400000*u^31)/( 29060388659606935628794638543460331673030534307223138660498676487084530156 19798986113305249885781959594801306131448969245977720589538127715431122138 44578745188503727328628116999363457247358791799197315263369261789993217556 480000*w_1^2) i16 : g3=toString(f0#3) o16 = (15552*w_2-69177612*w_1^6*u^39+248832*w_1^6*u^2-16941456*w_1^5*u^29- 4148928*w_1^4*u^19-1451520*w_1^3*u^9-69177612*w_1^2*u^36-16941456*w_1*u^26 +282475249*u^53+762048*u^16)/(w_1) i17 : g4=toString(f0#4) o17 = (-201684*w_1^6*u^28-49392*w_1^5*u^18-12096*w_1^4*u^8-201684*w_1^2*u^25- 49392*w_1*u^15+823543*u^42-6912*u^5)/(w_1) i18 : g5=toString(f0#5) o18 = (28*w_1^4*u^9-49*w_1*u^16+16*u^6)/(w_1) i19 : g6=toString(f0#6) o19 = (-w)/(u) i20 : g7=toString(f0#7) o20 = (w)/(y)