# Singular

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Topic review - vanishing points of a homogeneous ideal
Author Message
 steenpass
 Post subject: Re: vanishing points of a homogeneous ideal
 The zero set of this ideal is two-dimensional:Code:> ring r = 0, (w,x,y,z), dp;> ideal I =  wxy + x2y + xy2 + xyz, w2y + wxy + wy2 + wyz, w2x + wx2 + wxy + wxz, wxy;> I = std(I);> I;I[1]=x2y+xy2+xyzI[2]=wxyI[3]=w2y+wy2+wyzI[4]=w2x+wx2+wxz> dim(I);2Given the above standard basis, it's relatively easy to compute the vanishing set 'by hand'. If not, then computing a primary decomposition might help:Code:> LIB "primdec.lib";[snip]> primdecGTZ(I);[1]:   [1]:      _[1]=x      _[2]=w+y+z   [2]:      _[1]=x      _[2]=w+y+z[2]:   [1]:      _[1]=y      _[2]=w+x+z   [2]:      _[1]=y      _[2]=w+x+z[3]:   [1]:      _[1]=y      _[2]=x   [2]:      _[1]=y      _[2]=x[4]:   [1]:      _[1]=x+y+z      _[2]=w   [2]:      _[1]=x+y+z      _[2]=w[5]:   [1]:      _[1]=y      _[2]=w   [2]:      _[1]=y      _[2]=w[6]:   [1]:      _[1]=x      _[2]=w   [2]:      _[1]=x      _[2]=w The zero set of this ideal is two-dimensional:[code]> ring r = 0, (w,x,y,z), dp;> ideal I = wxy + x2y + xy2 + xyz, w2y + wxy + wy2 + wyz, w2x + wx2 + wxy + wxz, wxy;> I = std(I);> I;I[1]=x2y+xy2+xyzI[2]=wxyI[3]=w2y+wy2+wyzI[4]=w2x+wx2+wxz> dim(I);2[/code]Given the above standard basis, it's relatively easy to compute the vanishing set 'by hand'. If not, then computing a primary decomposition might help:[code]> LIB "primdec.lib";[snip]> primdecGTZ(I);[1]: [1]: _[1]=x _[2]=w+y+z [2]: _[1]=x _[2]=w+y+z[2]: [1]: _[1]=y _[2]=w+x+z [2]: _[1]=y _[2]=w+x+z[3]: [1]: _[1]=y _[2]=x [2]: _[1]=y _[2]=x[4]: [1]: _[1]=x+y+z _[2]=w [2]: _[1]=x+y+z _[2]=w[5]: [1]: _[1]=y _[2]=w [2]: _[1]=y _[2]=w[6]: [1]: _[1]=x _[2]=w [2]: _[1]=x _[2]=w[/code]
 Posted: Thu Jan 12, 2017 9:25 am
 ibahmani
 Post subject: vanishing points of a homogeneous ideal
 I am trying to find vanishing points of a homogeneous idealI tried to use Singular to find but it seems there is not any function. Is there anyone who knows to do it? Has anyone tried it?I=(wxy + x^2y + xy^2 + xyz, w^2y + wxy + wy^2 + wyz, w^2x + wx^2 + wxy + wxz, wxy) I am trying to find vanishing points of a homogeneous idealI tried to use Singular to find but it seems there is not any function. Is there anyone who knows to do it? Has anyone tried it?I=(wxy + x^2y + xy^2 + xyz, w^2y + wxy + wy^2 + wyz, w^2x + wx^2 + wxy + wxz, wxy)
 Posted: Wed Jan 11, 2017 7:28 pm

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