Singular
https://www.singular.uni-kl.de/forum/

Ideal from set of points
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1580		 Page 1 of 1
Author:  Azumanga [ Mon Dec 04, 2006 6:07 pm ]
Post subject:  Ideal from set of points
I'm sorry if this is a FAQ, but is there a easy way in Singular to generate an ideal from a set of points. I mean I want the ideal whose solutions are <x,y> in { <1,1>,<2,3>,<3,2> }. Typically I have up to 50 points.
Author:  hannes [ Wed Dec 06, 2006 3:07 pm ]
Post subject:  Re: Ideal from set of point (with mulitplicities)
Singular 3-0-2-1 from Oct 9 2006 (and newer) has the new command interpolation for exactly this purpose:

interpolation
-------------

`*Syntax:*'
`interpolation (' list`,' intvec `)'
`*Type:*'
ideal
`*Purpose:*'
`interpolation(l,v)' computes the reduced Groebner basis of the
intersection of ideals l[1]^v[1],...,l[N]^v[N] by applying linear
algebra methods.
`*Assume:*'
Every ideal from the list l must be a maximal ideal of a point and
should have the following form:
variable_1-coordinate_1,...,variable_n-coordinate_n, where n is
the number of variables in the ring.
The ring should be a polynomial ring over Zp or Q with global
ordering.
`*Example:*'
ring r=0,(x,y),dp;
ideal p_1=x,y;
ideal p_2=x+1,y+1;
ideal p_3=x+2,y-1;
ideal p_4=x-1,y+2;
ideal p_5=x-1,y-3;
ideal p_6=x,y+3;
ideal p_7=x+2,y;
list l=p_1,p_2,p_3,p_4,p_5,p_6,p_7;
intvec v=2,1,1,1,1,1,1;
ideal j=interpolation(l,v);
// generator of degree 3 gives the equation of the unique
// singular cubic passing
// through p_1,...,p_7 with singularity at p_1
j;
==> j[1]=-4x3-4x2y-2xy2+y3-8x2-4xy+3y2
==> j[2]=-y4+8x2y+6xy2-2y3+10xy+3y2
==> j[3]=-xy3+2x2y+xy2+4xy
==> j[4]=-2x2y2-2x2y-2xy2+y3-4xy+3y2
// computes values of generators of j at p_4, results should be 0
subst(j,x,1,y,-2);
==> _[1]=0
==> _[2]=0
==> _[3]=0
==> _[4]=0
// computes values of derivatives d/dx of generators at (0,0)
subst(diff(j,x),x,0,y,0);
==> _[1]=0
==> _[2]=0
==> _[3]=0
==> _[4]=0
Page 1 of 1 All times are UTC + 1 hour [ DST ]
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group
http://www.phpbb.com/