testElement -- find a test element of a ring

Synopsis

Usage:

testElement(R)
Inputs:
- R, a ring, a quotient of a polynomial ring
Optional inputs:
- AssumeDomain => a Boolean value, default value false, assumes the ring is a domain
Outputs:
- a ring element, a test element of R

Description

Given $R = S/I$, where $S$ is a polynomial ring, the function testElement finds an element of the ambient ring $S$ whose image in $R$ is a test element of $R$. This is done by finding a minor of the jacobian of $I$ that does not lie in any minimal prime of $I$. This function considers random minors until one is found, instead of computing all minors. Thus, repeated calls will not always produce the same answer.

i1 : R = ZZ/11[x,y,z]/(x^3 + y^3 + z^3);

i2 : apply(1..5, i -> testElement(R))

         2     2    2    2    2
o2 = (-2y , -2z , -y , 2y , 4x )

o2 : Sequence

If the option AssumeDomain (default value false) is set to true, then testElement does not compute the minimal primes of $I$. This can result in a substantial speedup in some cases.

Ways to use testElement :

testElement(Ring)

For the programmer

The object testElement is a method function with options.