eulerOperators -- Euler Operators

Synopsis

Usage:

eulerOperators(A, D)

eulerOperators(A, b, D)
Inputs:
- A, a matrix,
- b, a list,
- D, a polynomial ring,
Outputs:
- a list, of Euler operators

Description

Given a $d \times n$ integer matrix $A = (a_{ij})$ and a Weyl algebra in $n$ variables, produce the $d$ corresponding Euler operators $E_i = \sum_{j=1}^n a_{ij}x_jdj$. An optional list $b$ imposes a multigrading so that one can look for solutions to the Euler operatros of multidegree $b$.

i1 : D = makeWeylAlgebra(QQ[x,y,z])

o1 = D

o1 : PolynomialRing, 3 differential variable(s)

i2 : A = matrix{{2,-7,5},{14,8,-1}}

o2 = | 2  -7 5  |
     | 14 8  -1 |

              2       3
o2 : Matrix ZZ  <-- ZZ

i3 : L = eulerOperators(A,D)

o3 = {2x*dx - 7y*dy + 5z*dz, 14x*dx + 8y*dy - z*dz}

o3 : List

i4 : Example

o4 = Example

o4 : Symbol

i5 : D = makeWeylAlgebra(QQ[x,y,z])

o5 = D

o5 : PolynomialRing, 3 differential variable(s)

i6 : A = matrix{{2,-7,5},{14,8,-1}}

o6 = | 2  -7 5  |
     | 14 8  -1 |

              2       3
o6 : Matrix ZZ  <-- ZZ

i7 : b = {2,-3}

o7 = {2, -3}

o7 : List

i8 : L = eulerOperators(A,b,D)

o8 = {2x*dx - 7y*dy + 5z*dz - 2, 14x*dx + 8y*dy - z*dz + 3}

o8 : List

Caveat

Ring input should be a Weyl algebra. Matrix input should have as many columns as variables of the Weyl algebra. List should have as many entries as there are rows of matrix.

Ways to use eulerOperators :

eulerOperators(Matrix,List,PolynomialRing)
eulerOperators(Matrix,PolynomialRing)

For the programmer