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 |