Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ where $f_n = g + h$. Suppose the sets $\{ f_1,...,f_{n-1},g \}$ and $\{ f_1,...,f_{n-1},h \}$ are algebraically independent then is there a characterisation of when $\{f_1,...,f_n\}$ is algebraically independent. Ideally, I want to know if an efficient algorithm exists to test this. An example where algebraic independence is not preserved is :
$f_1 = x_1 + x_2^2$
$f_2 = x_1^2 + x_2^4$
when $\mathbb{F} = \mathbb{F}_2$ and $g = x_1^2$ and $h = x_2^4$.
