# DependentSet -- option for computing Noetherian operators

## Description

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

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.

## Caveat

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