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coordinateChangeOps -- induced Noetherian operators under coordinate change

Synopsis

Description

Let $I$ be an ideal in a polynomial ring $K[x_1, ..., x_n]$, and $\phi \in GL_n(K)$ a matrix representing a $K$-linear automorphism of $R$. Then there is an automorphism $\psi$ of the Weyl algebra $K[x_i, dx_i]$ such that if $D_1, ..., D_r$ is a set of Noetherian operators for $I$ then $\psi(D_1), ..., \psi(D_r)$ is a set of Noetherian operators for $\phi(I)$. This function computes the induced operators for a given $\phi$. The action of $\psi$ on polynomial variables $x_i$ is given by $\phi$, while the action of $\psi$ on differential variables $dx_i$ is given by the inverse transpose of $\phi$.

i1 : R = QQ[x,y,t]

o1 = R

o1 : PolynomialRing
 
i2 : I = ideal(x^2, y^2 - x*t)

             2   2
o2 = ideal (x , y  - x*t)

o2 : Ideal of R
 
i3 : P = radical I

o3 = ideal (y, x)

o3 : Ideal of R
 
i4 : N = noetherianOperators I

o4 = {| 1 |, | dy |, | tdy^2+2dx |, | tdy^3+6dxdy |}

o4 : List
 
i5 : phi = map(R, R, diagonalMatrix apply(numgens R, i -> random QQ))

                 9   1   9
o5 = map (R, R, {-x, -y, -t})
                 2   2   4

o5 : RingMap R <-- R
 
i6 : N' = coordinateChangeOps_phi N

o6 = {| 1 |, | 2dy |, | 4tdy^2+4/9dx |, | 8tdy^3+8/3dxdy |}

o6 : List
 
i7 : I' = phi I

            81 2  1 2   81
o7 = ideal (--x , -y  - --x*t)
             4    4      8

o7 : Ideal of R
 
i8 : P' = phi P

            1   9
o8 = ideal (-y, -x)
            2   2

o8 : Ideal of R
 
i9 : I' == getIdealFromNoetherianOperators(N', P')

o9 = false

See also

Ways to use coordinateChangeOps :

For the programmer

The object coordinateChangeOps is a method function.