Let $k$ be a field, $A$ a finitely generated $k$-algebra. By Noether normalization, we know that there exists a finite morphism of $k$-algebras $\varphi : k[x_1, \ldots, x_d] \hookrightarrow A$, with $d = \dim A$.
I say that a tuple $u_1, \ldots, u_d$ satisfies Noether normalization, if the elements are algebraically independent over $k$, and the (injective) morphism $\varphi_\boldsymbol{u} : k[x_1, \ldots, x_d] \hookrightarrow A$, $x_i \mapsto u_i$ is a finite morphism.
My question: given a tuple $a_1, \ldots, a_d \in A$, which form an algebraically independent set over $k$, is there an easy way to see if they satisfy Noether normalization?
More generally, is there a construction, that, given $a_1, \ldots, a_d$ algebraically independent, produces $b_1, \ldots, b_d$ which satisfy Noether normalization?
Note: Choosing the algebraic independent elements in Noether's normalization lemma is a related question, but does not quite answer mine.
