Let $K$ be an algebraically closed field with valuation, and its value group $\Gamma_\text{val}^n$ is dense in $\mathbb{R}$. Consider the polynomial $ f(x,y)=x+y+1. $ It can easy by shown that the tropical hypersurface $V(\text{trop}(f))$ is the standard tropical lines $ V(\text{trop}(f))=\{x=y \leq 0\} \cup \{x=0 \leq y\} \cup \{y=0\leq x\}. $

I want to show that this is true from just taking pointwise valuation. We have that $ V(f)=\{(z,-1-1)\}. $

note that $\text{in}_\mathbf{w}(f)$ is a monomial unless $\mathbf{w}$ is a non negative multipe of $(1,0),(0,1)$ or $(1,1)$. My problem becomes in showing/understanding that

$ (\text{val}(z),\text{val}(-1-z)= \begin{cases} (\text{val}(z),0), \ \text{if} \ \text{val}(z)>0 \\ (\text{val}(z),\text{val}(z),\ \text{val}(z) <0 \\ (0,\text{val}(z+1), \ \text{if} \ \text{val}(z)=0, \ \text{val}(z+1)>0\\ 0, \quad \text{otherwise} \end{cases} $

is correct. I usually think of $K$ to be the Puiseux series.


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