next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
TriangularSets :: RingElement % TriaSystem

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

Synopsis

Description

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

Let T = (t1,t2,…,tk) where mvar(t1)>…>mvar(tk). The pseudo-remainder of f by T is

prem(f,T) = prem(…(prem(prem(f,t1),t2)…,tk)

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

See also