# symmetricKernel(..., Variable => ...) -- Choose name for variables in the created ring

## Synopsis

• Usage:
symmetricKernel(...,Variable=>w)
reesIdeal(...,Variable=>w)
reesAlgebra(...,Variable=>w)
normalCone(...,Variable=>w)
specialFiberIdeal(...,Variable=>w)
specialFiber(...,Variable=>w)
distinguished(...,Variable=>w)
distinguishedAndMult(...,Variable=>w)

## Description

Each of these functions creates a new ring of the form R[w0, ..., wr] or R[w0, ..., wr]/J, where R is the ring of the input ideal or module (except for specialFiber, which creates a ring K[w0, ..., wr], where K is the ultimate coefficient ring of the input ideal or module.) This option allows the user to change the names of the new variables in this ring. The default variable is w.
 ```i1 : R = QQ[x,y,z]/ideal(x*y^2-z^9) o1 = R o1 : QuotientRing``` ```i2 : J = ideal(x,y,z) o2 = ideal (x, y, z) o2 : Ideal of R``` ```i3 : I = reesIdeal(J, Variable => p) 8 2 o3 = ideal (z*p - y*p , z*p - x*p , y*p - x*p , x*y*p - z p , x*p - 1 2 0 2 0 1 1 2 1 ------------------------------------------------------------------------ 7 2 2 6 3 z p , p p - z p ) 2 0 1 2 o3 : Ideal of R[p , p , p ] 0 1 2```
To lift the result to an ideal in a flattened ring, use flattenRing:
 ```i4 : describe ring I o4 = R[p , p , p , Degrees => {3:{1}}, Heft => {1, 0}, MonomialOrder => 0 1 2 {1} ------------------------------------------------------------------------ {MonomialSize => 32}, DegreeRank => 2] {GRevLex => {3:1} } {Position => Up }``` ```i5 : I1 = first flattenRing I 9 2 8 2 o5 = ideal (- z + x*y , p z - p y, p z - p x, p y - p x, p x*y - p z , p x - 1 2 0 2 0 1 1 2 1 ------------------------------------------------------------------------ 2 7 2 3 6 p z , p p - p z ) 2 0 1 2 o5 : Ideal of QQ[p , p , p , x, y, z] 0 1 2``` ```i6 : describe ring oo o6 = QQ[p , p , p , x..z, Degrees => {3:{1}, 3:{0}}, Heft => {0..1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 2] 0 1 2 {1} {1} {GRevLex => {3:1} } {Position => Up } {GRevLex => {3:1} }```
Note that the rings of I and I1 both have bigradings. Use newRing to make a new ring with different degrees.
 ```i7 : S = newRing(ring I1, Degrees=>{numgens ring I1:1}) o7 = S o7 : PolynomialRing``` ```i8 : describe S o8 = QQ[p , p , p , x..z, Degrees => {6:1}, Heft => {1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1] 0 1 2 {GRevLex => {3:1} } {Position => Up } {GRevLex => {3:1} }``` ```i9 : I2 = sub(I1,vars S) 9 2 8 2 o9 = ideal (- z + x*y , p z - p y, p z - p x, p y - p x, p x*y - p z , p x - 1 2 0 2 0 1 1 2 1 ------------------------------------------------------------------------ 2 7 2 3 6 p z , p p - p z ) 2 0 1 2 o9 : Ideal of S``` ```i10 : res I2 1 7 11 6 1 o10 = S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 o10 : ChainComplex```

## Further information

• Default value: w
• Function: symmetricKernel -- Compute the Rees ring of the image of a matrix
• Option name: Variable -- specify a name for a variable