# eulerOperators -- Euler Operators

## Synopsis

• Usage:
eulerOperators(A, D)
eulerOperators(A, b, D)
• Inputs:
• A, ,
• b, a list,
• D, ,
• 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.