Let $Q \subseteq R := \mathbb{K}[x_1,\dots,x_n]$ be an $d$-dimensional primary ideal. Then there exists a set of $d$ variables in $R$ which is algebraically independent in $R/I$. We refer to these as the independent variables, and the remaining variables are the dependent variables. The function independentSets can compute sets of independent variables for symbolic ideals.

The functions computing Noetherian operators, namely

- noetherianOperators -- Noetherian operators
- specializedNoetherianOperators -- Noetherian operators evaluated at a point
- numericalNoetherianOperators -- Noetherian operators via numerical interpolation

pass to a polynomial ring in the dependent variables, with the coefficient field being the fraction field of a polynomial ring in the independent variables. Because of this, computing Noetherian operators requires a knowledge of a dependent set of variables, which can be set using the option `DependentSet`. Note that the $dx$-monomials will only involve dependent variables.

i1 : R = QQ[x,y]; |

i2 : I = ideal((x+y)^2); o2 : Ideal of R |

i3 : P = radical I; o3 : Ideal of R |

i4 : A = noetherianOperators(I, P, DependentSet => {x}) o4 = {| 1 |, | dx |} o4 : List |

i5 : B = noetherianOperators(I, P, DependentSet => {y}) o5 = {| 1 |, | dy |} o5 : List |

i6 : getIdealFromNoetherianOperators(A, P) == getIdealFromNoetherianOperators(B, P) o6 = true |

The symbolic method noetherianOperators will usually be able to figure out a dependent set of variables automatically. On the other hand, numerical computations using specializedNoetherianOperators and numericalNoetherianOperators will usually require the user to set the option `DependentSet`.

The option `DependentSet` is ignored when calling noetherianOperators with Strategy => "PunctualQuot". Note that this is the default strategy for noetherianOperators(Ideal).

- noetherianOperators -- Noetherian operators
- specializedNoetherianOperators -- Noetherian operators evaluated at a point
- numericalNoetherianOperators -- Noetherian operators via numerical interpolation

The object DependentSet is a symbol.