Original Formulation of Hilbert's 14th Problem

Solution 1:

The two formulations are indeed equivalent.

The map $f: k(X_1,...,X_m) \rightarrow k(x_1,...,x_n)$ that sends $X_i$ to $f_i$ is an injection of fields. It identifies $k(X_1,...,X_m)$ with a subfield $K$ of $k(x_1,...,x_n)$. The intersection $R=K\cap k[x_1,...,x_n]$ is then all elements of $k(X_1,..,X_m)$ that are mapped to $k[x_1,...,x_n]$ via $f$, i.e. exactly the relatinganz functions (relative to f).

It's a fact that any subfield $K$ of $k(x_1,...,x_n)$ is finitely generated. To recover Hilbert's formulation, we only need to choose generators $f_1,...,f_m$ for $K$ (over $k$). Note that the notion of being relativganz depends only on the field $K$ generated by the $f_i$, not the individual generators.