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

## Synopsis

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

## Description

Each of these functions creates a new ring of the form R[w_0,\ldots, w_r] or R[w_0,\ldots, w_r]/J, where R is the ring of the input ideal or module (except for specialFiber, which creates a ring $K[w_0,\ldots, w_r]$, 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 7 2 o3 = ideal (x*p - y*p , y*p - z*p , x*p - z*p , z p - y p , z p - 1 2 0 1 0 2 0 2 0 ------------------------------------------------------------------------ 6 3 2 y*p p , z p - p p ) 1 2 0 1 2 o3 : Ideal of R[p ..p ] 0 2

To lift the result to an ideal in a flattened ring, use flattenRing:

 i4 : describe ring I o4 = R[p ..p , Degrees => {3:{1}}, Heft => {1, 0}, Join => false] 0 2 {1} i5 : I1 = first flattenRing I 9 2 8 2 2 7 o5 = ideal (- z + x*y , p x - p y, p y - p z, p x - p z, p z - p y , p z - 1 2 0 1 0 2 0 2 0 ------------------------------------------------------------------------ 3 6 2 p p y, p z - p p ) 1 2 0 1 2 o5 : Ideal of QQ[p ..p , x..z] 0 2 i6 : describe ring oo o6 = QQ[p ..p , x..z, Degrees => {3:{1}, 3:{0}}, Heft => {0..1}, Join => false, MonomialOrder => {MonomialSize => 32}] 0 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 , x..z, Degrees => {6:1}, Heft => {1}, Join => false, MonomialOrder => {MonomialSize => 32}] 0 2 {GRevLex => {3:1} } {Position => Up } {GRevLex => {3:1} } i9 : I2 = sub(I1,vars S) 9 2 8 2 2 7 o9 = ideal (- z + x*y , p x - p y, p y - p z, p x - p z, p z - p y , p z - 1 2 0 1 0 2 0 2 0 ------------------------------------------------------------------------ 3 6 2 p p y, p z - p p ) 1 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 key: Variable -- specify a name for a variable

• flattenRing -- write a ring as a (quotient of a) polynomial ring
• newRing -- make a copy of a ring, with some features changed
• substitute -- substituting values for variables

## Functions with optional argument named Variable :

• conwayPolynomial(...,Variable=>...) (missing documentation)
• "GF(...,Variable=>...)" -- see GF -- make a finite field
• "Grassmannian(...,Variable=>...)" -- see Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
• idealizer(...,Variable=>...) -- Sets the name of the indexed variables introduced in computing the endomorphism ring Hom(J,J).
• integralClosure(...,Variable=>...) -- set the base letter for the indexed variables introduced while computing the integral closure
• "intersectInP(...,Variable=>...)" -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
• "multiplicity(...,Variable=>...)" -- see intersectInP(...,BasisElementLimit=>...) -- Option for intersectInP
• makeS2(...,Variable=>...) -- Sets the name of the indexed variables introduced in computing the S2-ification.
• "ringFromFractions(...,Variable=>...)" -- see ringFromFractions -- find presentation for f.g. ring
• "Schubert(...,Variable=>...)" -- see Schubert(ZZ,ZZ,VisibleList) -- find the Plücker ideal of a Schubert variety
• "distinguished(...,Variable=>...)"
• "isReduction(...,Variable=>...)"
• "jacobianDual(...,Variable=>...)"
• "normalCone(...,Variable=>...)"
• "reesAlgebra(...,Variable=>...)"
• "reesIdeal(...,Variable=>...)"
• "specialFiber(...,Variable=>...)"
• "specialFiberIdeal(...,Variable=>...)"
• symmetricKernel(...,Variable=>...) -- Choose name for variables in the created ring