I am trying to understand the proof of the fundamental theorem of tropical algebraic geometry from Maclagan-Sturmfels (Introduction to Tropical Geometry, Section 3.2 of the 2015 edition). Is the following an error in the proof of Proposition 3.2.11? It is not included in the errata (as of today), so wanted to make sure if my understanding is correct.
My concern is regarding the third and fourth paragraphs of the proof of Proposition 3.2.11 (in the 2015 edition). On the second paragraph a map $\phi$ is constructed, and on the third paragraph one picks a certain $f \in I$. Then there are claims in the third and fourth paragraphs that $f$ has a specific form due to the way $\phi$ was constructed. However, as far as I can understand, this argument is not valid since the construction of $\phi$ (using Proposition 3.2.7) guarantees that particular form for a polynomial only if it had been picked before $\phi$ is constructed, whereas our $f$ depends on $\phi$. What am I missing?
In case it is indeed an error, it would be great to see ideas on how to fix it. Here is one way I think it can be done: $f(y_1, \ldots, y_{n-1},x_n)$ is the $\gcd$ of a set of generators $f_1, \ldots, f_s$ of $I$ (which can be fixed a priori), and therefore $f = \sum_i r_if_i$, where the $r_i$ are rational functions in $y' = (y_1, \ldots, y_{n-1})$. Then one can use the inductive hypothesis that the set of $y'$ is dense in $X'$ to show that $in_{w_n}(f(y',x_n))$ is not a monomial for generic $y'$, which suffices to complete the proof.
