# coordinateChangeOps -- induced Noetherian operators under coordinate change

## Synopsis

• Usage:
coordinateChangeOps(phi, D)
coordinateChangeOps(phi, L)
• Inputs:
• Outputs:
• , (resp. a list of differential operators)

## 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 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

## Ways to use coordinateChangeOps :

• "coordinateChangeOps(Matrix,DiffOp)"
• "coordinateChangeOps(Matrix,List)"
• "coordinateChangeOps(RingMap,DiffOp)"
• "coordinateChangeOps(RingMap,List)"

## For the programmer

The object coordinateChangeOps is .