# definingIdeal -- presentation of a ring of invariants as polynomial ring modulo the defining ideal

## Synopsis

• Usage:
definingIdeal S
• Inputs:
• Optional inputs:
• Variable (missing documentation) => ..., default value "u", name of the variables in the polynomial ring.
• Outputs:
• an ideal, which defines the ring of invariants as a polynomial ring modulo the ideal.

## Description

This function is provided by the package InvariantRing.

This method presents the ring of invariants as a polynomial ring modulo the defining ideal. The default variable name in the polynomial ring is u_i. You can pass the variable name you want as optional input.

 i1 : R = QQ[x_1..x_4] o1 = R o1 : PolynomialRing i2 : W = matrix{{0,1,-1,1},{1,0,-1,-1}} o2 = | 0 1 -1 1 | | 1 0 -1 -1 | 2 4 o2 : Matrix ZZ <--- ZZ i3 : T = diagonalAction(W, R) * 2 o3 = R <- (QQ ) via | 0 1 -1 1 | | 1 0 -1 -1 | o3 : DiagonalAction i4 : S = R^T o4 = 2 QQ[x x x , x x x ] 1 2 3 1 3 4 o4 : RingOfInvariants i5 : definingIdeal S o5 = ideal () o5 : Ideal of QQ[u ..u ] 1 2

## Ways to use definingIdeal :

• "definingIdeal(RingOfInvariants)"

## For the programmer

The object definingIdeal is .