RegularChains[MatrixTools]
MatrixOverChain
normal form of a matrix with respect to a regular chain
Calling Sequence
Parameters
Description
Examples
MatrixOverChain(A, rc, R)
A
-
Matrix with coefficients in the field of fractions of R
rc
regular chain of R
R
polynomial ring
The command MatrixOverChain(A, rc, R) returns the normal form of A with respect to rc. In broad terms, this is obtained by mapping RegularChains[NormalForm] on the coefficients of A.
The result is viewed as a matrix with coefficients in the total ring of fractions of R/I where I is the saturated ideal of rc.
It is assumed that rc is strongly normalized.
This command is part of the RegularChains[MatrixTools] package, so it can be used in the form MatrixOverChain(..) only after executing the command with(RegularChains[MatrixTools]). However, it can always be accessed through the long form of the command by using RegularChains[MatrixTools][MatrixOverChain](..).
withRegularChains:withChainTools:withMatrixTools:
R≔PolynomialRingx,y,z
R≔polynomial_ring
T≔EmptyR:
T≔Chainz+1z+2,y2+z,x−zx−y,T,R
T≔regular_chain
EquationsT,R
x2+−y−zx+zy,y2+z,z2+3z+2
m≔Matrixx,y,z,x2,y2,z2,x3,y5,z6
m≔xyzx2y2z2x3y5z6
MatrixOverChainm,T,R
xyzxy+zx−zy−z−3z−2xyz−4zx+3zy−2x+2y−3z−2−3zy−2y−63z−62,regular_chain
See Also
Chain
Empty
Equations
IsStronglyNormalized
IsZeroMatrix
JacobianMatrix
LowerEchelonForm
Matrix
MatrixInverse
MatrixMultiply
MatrixTools
NormalForm
PolynomialRing
RegularChains
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