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reduceDimension -- removes arcs of a chordal network of small dimension

Synopsis

Description

Reduces a chordal network, by prunning the arcs that do not belong to the top $k$ dimensional parts of the variety.

i1 : R = QQ[a..j,MonomialOrder=>Lex];
 
i2 : I = ideal {a*d - b*c, c*f - d*e, e*h - f*g, g*j - h*i, a*j - b*i};

o2 : Ideal of R
 
i3 : N = chordalNet I;
 
i4 : chordalTria N;
 
i5 : codimCount N

       7      6      5    4
o5 = 2t  + 17t  + 14t  + t

o5 : ZZ[t]
 
i6 : reduceDimension(N,2);
 
i7 : codimCount N

        5    4
o7 = 14t  + t

o7 : ZZ[t]
 
i8 : reduceDimension(N,1);
 
i9 : codimCount N

      4
o9 = t

o9 : ZZ[t]

The method topComponents(ChordalNet) is equivalent to reduceDimension(N,1).


      

      

        
Caveat

        

          
An arc is removed only if all the chains through it have small dimension. If an arc belongs to a chain of large dimenson and a chain of small dimension, then the arc will not be removed.

        

      

      

        
See also

        
      

      

        
For the programmer

        
The object reduceDimension is a function closure.