# specializedNoetherianOperators -- Noetherian operators evaluated at a point

## Synopsis

• Usage:
specializedNoetherianOperators(I, pt)
• Inputs:
• Outputs:

## Description

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. 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 2 3 o7 = {1, dy, dy - dx, dy - 3*dx*dy} o7 : List i8 : B = noetherianOperators(I, P) / (D -> evaluate(D, pt)) / normalize 2 3 o8 = {1, dy, dy - dx, dy - 3*dx*dy} 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}) 2 3 o12 = {1, dy, .5*dy - .47619*dx, .166667*dy - .47619*dx*dy} o12 : List

## Caveat

It is assumed that the point lies on the variety of I

## Ways to use specializedNoetherianOperators :

• "specializedNoetherianOperators(Ideal,Matrix)"
• "specializedNoetherianOperators(Ideal,Point)"

## For the programmer

The object specializedNoetherianOperators is .