RingElement // Subring -- Subduction quotient with respect to a subring

Synopsis

Operator: //
Usage:

r = f // S
Inputs:
- f, a ring element,
- S, a subring,
Outputs:
- r, a ring element, a subduction quotient of f with respect to the subring

Description

The result is a ring element that lies in the presentationRing that has one variable for each generator of the subring $S$. A subduction quotient is an expression of $f - (f\%S) \in S$ 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 = 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

This function calls groebnerSubductionQuotient.

Ways to use this method: