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NoetherianOperators :: coordinateChangeOps

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

                 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

See also

Ways to use coordinateChangeOps :

For the programmer

The object coordinateChangeOps is a method function.