Numerically computes evaluations of Noetherian operators. If the point p lies on the variety of the minimal prime $P$, the function returns a set of specialized Noetherian operators of the $P$-primary component of $I$. The option DependentSet is required when dealing with ideals over numerical fields, or when dealing with non-primary ideals.
i1 : R = QQ[x,y,t]; |
i2 : Q1 = ideal(x^2, y^2 + x*t); o2 : Ideal of R |
i3 : Q2 = ideal((x+t)^2); o3 : Ideal of R |
i4 : I = intersect(Q1, Q2); o4 : Ideal of R |
i5 : P = radical Q1; o5 : Ideal of R |
i6 : pt = point{{0,0,2}};
|
i7 : A = specializedNoetherianOperators(I, pt, DependentSet => {x,y}) / normalize
o7 = {| 1 |, | dy |, | dy^2-dx |, | dy^3-3dxdy |}
o7 : List
|
i8 : B = noetherianOperators(I, P) /
(D -> evaluate(D,pt)) /
normalize
o8 = {| 1 |, | dy |, | dy^2-dx |, | dy^3-3dxdy |}
o8 : List
|
i9 : A == B o9 = true |
Over a non-exact field, the output will be non-exact
i10 : S = CC[x,y,t] o10 = S o10 : PolynomialRing |
i11 : pt = point{{0,0,2.1}}
o11 = pt
o11 : Point
|
i12 : specializedNoetherianOperators(sub(I, S), pt, DependentSet => {x,y})
o12 = {| 1 |, | dy |, | .5dy^2-.47619dx |, | .166667dy^3-.47619dxdy |}
o12 : List
|
It is assumed that the point lies on the variety of I
The object specializedNoetherianOperators is a method function with options.