# Singular

#### D.15.31.1 borderBasis

Procedure from library rstandard.lib (see rstandard_lib).

Usage:
borderBasis(I); I is an ideal.

Return:
ideal, a border basis for I.

Purpose:
Computes a border basis for the ideal given by the generators in I.

Example:
 LIB "rstandard.lib"; ring R=32003,(x,y,z),ds; poly f=x3y+x5+x3y2+2x2y3+x2yz2+xy5+x12+y16+z20; ideal i= jacob(f); i=i,f; ideal j=borderBasis(i); j; ==> j[1]=y4z20 ==> j[2]=y5z19 ==> j[3]=xy4z19 ==> j[4]=y6z18 ==> j[5]=xy5z18 ==> j[6]=y3z20 ==> j[7]=xy3z19 ==> j[8]=xy4z18 ==> j[9]=y6z17-12804y4z19 ==> j[10]=xy5z17 ==> j[11]=y7z16+16001y5z18 ==> j[12]=xy6z16 ==> j[13]=y8z15-6401y4z19 ==> j[14]=y9z14 ==> j[15]=xy8z14 ==> j[16]=z22 ==> j[17]=yz21-2461y4z19 ==> j[18]=xz21 ==> j[19]=y2z20 ==> j[20]=xyz20 ==> j[21]=xy2z19+y4z19 ==> j[22]=xy3z18+8001y5z18 ==> j[23]=xy4z17-12801y4z19 ==> j[24]=xy5z16 ==> j[25]=y7z15+16001y5z17+y3z19+6153y4z19 ==> j[26]=xy6z15 ==> j[27]=xy7z14 ==> j[28]=y9z13 ==> j[29]=xy8z13 ==> j[30]=xz20 ==> j[31]=xyz19+y3z19+6153y4z19 ==> j[32]=xy2z18-y8z14-16001y6z16 ==> j[33]=xy3z17+8001y5z17+16001y3z19+12925y4z19 ==> j[34]=xy4z16-16001y8z14 ==> j[35]=xy5z15 ==> j[36]=xy6z14 ==> j[37]=xy7z13 ==> j[38]=y9z12 ==> j[39]=xy8z12 ==> j[40]=xz19+y2z19+13336z21-10668y3z19-2051y4z19 ==> j[41]=xyz18-y7z14-16001y5z16 ==> j[42]=x2z18+15999y8z14 ==> j[43]=xy2z17-y8z13-16001y6z15 ==> j[44]=xy3z16-16001y7z14 ==> j[45]=xy4z15-16001y8z13 ==> j[46]=xy5z14 ==> j[47]=xy6z13 ==> j[48]=xy7z12 ==> j[49]=y9z11 ==> j[50]=xy8z11 ==> j[51]=xyz17-y7z13-16001y5z15 ==> j[52]=x2z17+15999y8z13 ==> j[53]=xy2z16-y8z12-16001y6z14 ==> j[54]=xy3z15-16001y7z13 ==> j[55]=xy4z14-16001y8z12 ==> j[56]=xy5z13 ==> j[57]=xy6z12 ==> j[58]=xy7z11 ==> j[59]=y9z10 ==> j[60]=xy8z10 ==> j[61]=xyz16-y7z12-16001y5z14 ==> j[62]=x2z16+15999y8z12 ==> j[63]=xy2z15-y8z11-16001y6z13 ==> j[64]=xy3z14-16001y7z12 ==> j[65]=xy4z13-16001y8z11 ==> j[66]=xy5z12 ==> j[67]=xy6z11 ==> j[68]=xy7z10 ==> j[69]=y9z9 ==> j[70]=xy8z9 ==> j[71]=xyz15-y7z11-16001y5z13 ==> j[72]=x2z15+15999y8z11 ==> j[73]=xy2z14-y8z10-16001y6z12 ==> j[74]=xy3z13-16001y7z11 ==> j[75]=xy4z12-16001y8z10 ==> j[76]=xy5z11 ==> j[77]=xy6z10 ==> j[78]=xy7z9 ==> j[79]=y9z8 ==> j[80]=xy8z8 ==> j[81]=xyz14-y7z10-16001y5z12 ==> j[82]=x2z14+15999y8z10 ==> j[83]=xy2z13-y8z9-16001y6z11 ==> j[84]=xy3z12-16001y7z10 ==> j[85]=xy4z11-16001y8z9 ==> j[86]=xy5z10 ==> j[87]=xy6z9 ==> j[88]=xy7z8 ==> j[89]=y9z7 ==> j[90]=xy8z7 ==> j[91]=xyz13-y7z9-16001y5z11 ==> j[92]=x2z13+15999y8z9 ==> j[93]=xy2z12-y8z8-16001y6z10 ==> j[94]=xy3z11-16001y7z9 ==> j[95]=xy4z10-16001y8z8 ==> j[96]=xy5z9 ==> j[97]=xy6z8 ==> j[98]=xy7z7 ==> j[99]=y9z6 ==> j[100]=xy8z6 ==> j[101]=xyz12-y7z8-16001y5z10 ==> j[102]=x2z12+15999y8z8 ==> j[103]=xy2z11-y8z7-16001y6z9 ==> j[104]=xy3z10-16001y7z8 ==> j[105]=xy4z9-16001y8z7 ==> j[106]=xy5z8 ==> j[107]=xy6z7 ==> j[108]=xy7z6 ==> j[109]=y9z5 ==> j[110]=xy8z5 ==> j[111]=xyz11-y7z7-16001y5z9 ==> j[112]=x2z11+15999y8z7 ==> j[113]=xy2z10-y8z6-16001y6z8 ==> j[114]=xy3z9-16001y7z7 ==> j[115]=xy4z8-16001y8z6 ==> j[116]=xy5z7 ==> j[117]=xy6z6 ==> j[118]=xy7z5 ==> j[119]=y9z4 ==> j[120]=xy8z4 ==> j[121]=xyz10-y7z6-16001y5z8 ==> j[122]=x2z10+15999y8z6 ==> j[123]=xy2z9-y8z5-16001y6z7 ==> j[124]=xy3z8-16001y7z6 ==> j[125]=xy4z7-16001y8z5 ==> j[126]=xy5z6 ==> j[127]=xy6z5 ==> j[128]=xy7z4 ==> j[129]=y9z3 ==> j[130]=xy8z3 ==> j[131]=xyz9-y7z5-16001y5z7 ==> j[132]=x2z9+15999y8z5 ==> j[133]=xy2z8-y8z4-16001y6z6 ==> j[134]=xy3z7-16001y7z5 ==> j[135]=xy4z6-16001y8z4 ==> j[136]=xy5z5 ==> j[137]=xy6z4 ==> j[138]=xy7z3 ==> j[139]=y9z2 ==> j[140]=xy8z2 ==> j[141]=xyz8-y7z4-16001y5z6 ==> j[142]=x2z8+15999y8z4 ==> j[143]=xy2z7-y8z3-16001y6z5 ==> j[144]=xy3z6-16001y7z4 ==> j[145]=xy4z5-16001y8z3 ==> j[146]=xy5z4 ==> j[147]=xy6z3 ==> j[148]=xy7z2 ==> j[149]=y9z+10y4z19 ==> j[150]=xy8z ==> j[151]=xyz7-y7z3-16001y5z5 ==> j[152]=x2z7+15999y8z3-10672z21-10671y3z19+11493y4z19 ==> j[153]=xy2z6-y8z2-16001y6z4 ==> j[154]=x2yz6 ==> j[155]=x3z6 ==> j[156]=xy3z5-16001y7z3 ==> j[157]=xy4z4-16001y8z2 ==> j[158]=xy5z3 ==> j[159]=xy6z2 ==> j[160]=xy7z-10y4z19 ==> j[161]=y9-13yz20 ==> j[162]=xy8 ==> j[163]=xyz6-y7z2-16001y5z4 ==> j[164]=xy2z5-y8z-16001y6z3-10y3z19-7333y4z19 ==> j[165]=x2yz5 ==> j[166]=x3z5+10672z21+10671y3z19-11493y4z19 ==> j[167]=xy3z4-16001y7z2 ==> j[168]=xy4z3-16001y8z+5y3z19-14791y4z19 ==> j[169]=xy5z2 ==> j[170]=xy6z-10y3z19+7388y4z19 ==> j[171]=xy7+13yz20 ==> j[172]=xyz5-y7z-16001y5z3-10y2z19+5333z21-10616y3z19+391y4z19 ==> j[173]=xy2z4-y8-16001y6z2-3x2z6-15994y8z2+13z20+15952yz20 ==> j[174]=x2yz4 ==> j[175]=x3z4+x2z6+15999y8z2 ==> j[176]=xy3z3-16001y7z+5y2z19+10666z21+10646y3z19-9017y4z19 ==> j[177]=x2y2z3-7393y4z19 ==> j[178]=xy4z2-16001y8-16000x2z6+7997y8z2+15995z20+8018yz20 ==> j[179]=xy5z-10y2z19-5333z21+10671y3z19-11493y4z19 ==> j[180]=xy6-3x2z6-15994y8z2+13z20+15yz20 ==> j[181]=x2yz3+10z21 ==> j[182]=x3z3+x2z5+15999y8z-25y3z19-9699y4z19 ==> j[183]=xy3z2-16001xyz4+8001y5z2+8002x2z6+11998y8z2+15994z20+15994yz20 ==> j[184]=x2y2z2+10yz20 ==> j[185]=xy4z-16001xy2z3+8001y6z+15994yz19+15994y2z19+15989y4z19 ==> j[186]=xy5-xyz4+y7+16001y5z2+15987x2z6+8037y8z2-15937z20-15931yz20 ==> j[187]=x2y4+3x2z6+15994y8z2-13z20+15983yz20 ==> j[188]=x2yz2+10z20 ==> j[189]=x3z2+x2z4+15999y8+15994x2z6-7982y8z2-15969z20-8147yz20 ==> j[190]=xy3z-16001xyz3+8001y5z+8002x2z5+11998y8z+15994z19+15994yz19+7957y3\ z19+14472y4z19 ==> j[191]=x2y2z+10yz19 ==> j[192]=x2y3+xyz4-y7-16001y5z2-15986x2z6+7962y8z2-68z20+15931yz20 ==> j[193]=x2yz+10z19 ==> j[194]=x3z+x2z3+15999xy2z3-8002y6z+15979yz19-15944y2z19+10696z21+10661y3z\ 19+7047y4z19 ==> j[195]=x2y2+10669xy4-10667xy2z2+10668y6-xyz4+y7+16001y5z2+15986x2z6-7962y\ 8z2+68z20+5426yz20 ==> j[196]=x2y+10669xy3-10667xyz2-10669xy4+10668y5+10667xy2z2-10668y6-10666x2\ z4+xyz4-y7-16001y5z2-5338y8-2x2z6+5y8z2-5326z20-13868yz20 ==> j[197]=x3+x2z2-3xy4-4xy2z2-2y6+8xyz4-8y7+4y5z2+122x2z6-305y8z2-507z20-694\ yz20