saturate(TriaSystem) -- saturated ideal of a triangular system

Synopsis

Function: saturate
Usage:

saturate T
Inputs:
- T, an instance of the type TriaSystem,
Optional inputs:
- BasisElementLimit => ..., default value infinity,
- DegreeLimit => ..., default value {},
- MinimalGenerators => ..., default value true,
- PairLimit => ..., default value infinity,
- Strategy => ..., default value null,
Outputs:
- an ideal, the saturated ideal of T

Description

Returns the saturated ideal of a triangular system.

Let $T = (F,H)$ be a triangular system. Denoting $J=ideal(F)$, the saturated ideal of $T$ is $$sat(T) = J : h^\infty = \{ f: h^k f \in J for some h\in H, k\in \mathbb{N} \}$$

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 : saturate T

o5 = 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)

o5 : Ideal of R

Ways to use this method: