RingElement % ChordalNet -- ideal membership test

Synopsis

Operator: %
Usage:

f % N

pseudoRemainder(f,N)
Inputs:
- f, a ring element,
- N, an instance of the type ChordalNet,
Outputs:
- a ring element, random linear combination of the pseudo-remainder of f by the chains of N

Description

This method gives a randomized algorithm for ideal membership. If $f$ lies in the saturated ideal of each of the chains of the network, then the output is always zero. Otherwise, it returns a nonzero element with high probability.

As an example, consider the ideal of cyclically adjacent minors.

i1 : I = adjacentMinorsIdeal(QQ,2,6);

o1 : Ideal of QQ[a..l]

i2 : X = gens ring I;

i3 : J = I + (X_0 * X_(-1) - X_1*X_(-2));

o3 : Ideal of QQ[a..l]

i4 : f = sum gbList J;

i5 : N = chordalNet J;

i6 : chordalTria N;

i7 : f % N == 0

o7 = true

Caveat

It is assumed that the base field has sufficiently many elements. For small finite fields one must work over a suitable field extension.

Ways to use this method: