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
2 3
o4 = {1, dy, t*dy + 2*dx, t*dy + 6*dx*dy}
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
2 4 3 8
o6 = {1, 2*dy, 9t*dy + -*dx, 18t*dy + -*dx*dy}
9 3
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 = true |
The object coordinateChangeOps is a method function.