Let $I$ be an unmixed ideal in a polynomial ring $R = K[x_1, ..., x_n]$, with primary decomposition $I = Q_1 \cap ... \cap Q_s$, where $Q_i$ is $P_i$-primary. If $N_i$ is a set of Noetherian operators for $Q_i$, then one can construct a set of differential operators $N$ for $I$ which satisfies the Noetherian operator condition: given $f \in R$, one has $f \in I$ iff $D(f) \in\sqrt{I}$ for all $D \in N$.
i1 : R = QQ[x,y,t] o1 = R o1 : PolynomialRing |
i2 : I = intersect(ideal((y+t)^2), ideal(x^2, y^2 - t*x))
4 2 3 2 2 2 3 2 2 2 2 2
o2 = ideal (y - x*y t + 2y t - 2x*y*t + y t - x*t , x y + 2x y*t + x t )
o2 : Ideal of R
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i3 : radI = radical I
2
o3 = ideal (y + y*t, x*y + x*t)
o3 : Ideal of R
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i4 : primes = associatedPrimes I
o4 = {ideal(y + t), ideal (y, x)}
o4 : List
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i5 : L = primes / (P -> (P, noetherianOperators(I, P)))
2 3
o5 = {(ideal(y + t), {1, dy}), (ideal (y, x), {1, dy, t*dy + 2*dx, t*dy +
------------------------------------------------------------------------
6*dx*dy})}
o5 : List
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i6 : N = noethOpsFromComponents L
2 2 2 3
o6 = {1, dy, (y*t + t )dy + (2y + 2t)dx, (y*t + t )dy + (6y + 6t)dx*dy}
o6 : List
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i7 : all(flatten table(N, I_*, (D, f) -> (D f) % radI == 0), identity) o7 = true |
Note that this construction justifies the focus of Noetherian operators on the case that the ideal I is primary: in order to get a useful membership test for a non-primary (but still unmixed) ideal, it suffices to compute Noetherian operators on each primary component, and then combine them in the way given above.
The object noethOpsFromComponents is a method function.