Based on [1, Section 9.5], it is known that for any multivariate polynomial over $\mathbb{R}$ with indeterminate $x$ and $y$ (take two variables for simplicity), denoted by $f(x,y)$, it holds that $\mathbb{R}^2 \setminus \mathcal{X}$ is open and dense in $\mathbb{R}^2$ (if $ \mathcal{X} \not= \mathbb{R}^2$), where $$ \mathcal{X} = \{[x,y]\in \mathbb{R}^2 | f(x,y) = 0\}. $$
The question is as follows.
Denote the field of rational functions as $\mathcal{V}$, and consider now a multivariate polynomial over the field of rational functions, i.e. $\mathcal{V}[x,y]$. Denote such polynomial as $g(x,y)$. Let $$ \bar{\mathcal{X}} = \{[x,y] \in \mathcal{V}^2 | g(x,y) = 0\}. $$
Does it holds true that $\mathcal{V}^2 \setminus \bar{\mathcal{X}}$ is open and dense in $\mathcal{V}^2$ when $\bar{\mathcal{X}} \not= \mathcal{V}^2$?
[1] J. W. Polderman and J. C. Willems, "Introduction to mathematical systems theory: a behavioral approach", Springer Science, 1997.