Let $K$ be a field- either $\mathbb{C}$ or a finite field $\mathbb{F}_q$.

Suppose $X_1,X_2,\dots,X_n$ are $n$ elements such that the extension field $$K(X_1,X_2,\dots,X_n)$$ has transcendence degree $k < n$ over $K$. In other words, there are some algebraic dependencies among the $n$ elements $X_1,X_2,\dots,X_n$.

Is there some way of obtaining $k$ elements $Y_1,Y_2,\dots,Y_k$ from $X_1,X_2,\dots,X_n$ so that $K(Y_1,Y_2,\dots,Y_k)$ has transcendence degree exactly $k$?

More precisely, I am asking if there exist some $k$ functions (or polynomials) $f_1,f_2,\dots,f_k$ that take $X_1,X_2,\dots,X_n$ and give us $Y_j=f_j(X_1,\dots,X_n)$ so that whenever $X_1,\dots,X_n$ have transcendence degree $k$, the $k$ new elements $Y_1,Y_2,\dots,Y_k$ have full transcendence degree. In some sense, we want to get rid of redundancies and extract only elements that are algebraically independent.

I don't know if this is always possible, but I'll be happy even with some pointers or references.