Post new topic Reply to topic  [ 2 posts ] 
Author Message
 Post subject: vanishing points of a homogeneous ideal
PostPosted: Wed Jan 11, 2017 7:28 pm 

Joined: Wed Jan 11, 2017 6:58 pm
Posts: 1
I am trying to find vanishing points of a homogeneous ideal
I 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)


Report this post
Top
 Profile  
Reply with quote  
 Post subject: Re: vanishing points of a homogeneous ideal
PostPosted: Thu Jan 12, 2017 9:25 am 

Joined: Thu Mar 04, 2010 1:29 pm
Posts: 14
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+xyz
I[2]=wxy
I[3]=w2y+wy2+wyz
I[4]=w2x+wx2+wxz
> dim(I);
2


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


Report this post
Top
 Profile  
Reply with quote  
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

You can post new topics in this forum
You can reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

It is currently Mon Dec 18, 2017 3:23 am
cron
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group