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# 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 => , default value false, assumes the ring is a domain
• Outputs:
• , 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 .