RingElement % TriaSystem -- pseudo-remainder by a triangular set

Synopsis

Operator: %
Usage:

f % T

pseudoRemainder(f,T)
Inputs:
- f, a ring element,
- T, an instance of the type TriaSystem,
Outputs:
- a ring element, pseudo-remainder of f by T

Description

Returns the pseudo-remainder, $prem(f,T)$, of $f$ by a triangular set $T$.

Let $T = (t_1,t_2,\cdots,t_k)$ where $mvar(t_1)>\cdots>mvar(t_k)$. The pseudo-remainder of $f$ by $T$ is $$prem(f,T) = prem(\cdots(prem(prem(f,t_1),t_2)\cdots,t_k)$$

Remark: If $T$ is a regular chain, then $f$ lies in its saturated ideal iff $prem(f,T)=0$.

i1 : R = QQ[a,b,c,d,e,f,g,h, MonomialOrder=>Lex];

i2 : F = {a*d - b*c, c*f - d*e, e*h - f*g};

i3 : H = {d, f, h};

i4 : T = triaSystem(R,F,H)

o4 = {a*d - b*c, c*f - d*e, e*h - f*g} / {d, f, h}

o4 : TriaSystem

i5 : (a*h - b*g) % T

o5 = 0

o5 : R

i6 : saturate T

o6 = ideal (e*h - f*g, c*h - d*g, c*f - d*e, a*h - b*g, a*f - b*e, a*d - b*c)

o6 : Ideal of R

Ways to use this method: