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groebnerSubductionQuotient -- Extrinsic method for subduction quotients

Synopsis

Description

Uses the extrinsic method to compute a subduction quotient of $f$ by the generators of $S$. A subduction quotient is an element of the presentationRing. The variables of this ring correspond to the generators of $S$. A subduction quotient is a polynomial representation of $f - f\%S$ in terms of the generators of $S$. The value of this element in the ambient(Subring) ring can be recovered using the field presentationMap of the subring.

i1 : R = QQ[x,y];
 
i2 : S = subring {x+y, x*y, x*y^2};
 
i3 : f = x^3;
 
i4 : f - (f%S)

      3    2        2    3
o4 = x  + x y + 2x*y  + y

o4 : R
 
i5 : gs = groebnerSubductionQuotient(f, S)

      3
o5 = p  - 2p p  + p
      0     0 1    2

o5 : QQ[p ..p ]
         0   2
 
i6 : S#"presentationMap" gs

      3    2        2    3
o6 = x  + x y + 2x*y  + y

o6 : R

See also

Ways to use groebnerSubductionQuotient:

For the programmer

The object groebnerSubductionQuotient is a method function.