diffOpRing -- create and cache the ring of differential operators

Synopsis

Takes a polynomial ring $R = \mathbb{K}[x_1,\dotsc,x_n]$ and creates the ring $S = R[dx_1,\dotsc,dx_n]$.

i1 : R = QQ[x_1..x_3, a,b];

i2 : S = diffOpRing R;

i3 : gens S

o3 = {dx , dx , dx , da, db}
        1    2    3

o3 : List

i4 : coefficientRing S

o4 = R

o4 : PolynomialRing

Differential operators on $R$ have entries in $S$.

i5 : ring diffOp(dx_3^2) === S

o5 = true

i6 : ring diffOp(a_R) === S

o6 = true

Subsequent calls to diffOpRing will not create new rings

i7 : diffOpRing R === S

o7 = true

the created ring is not a Weyl algebra, it is a commutative ring