Tropical hypersurface definition (book picture included).

Question

I don't entirely get how this is picture is the locus of points where the function is not linear.

I don't get what function is supposed to not be linear.

I can take any help that I can get right now thanks .

Answer 1

Observe that $0\oplus(0\odot x_1)=\max\{0,0+x_1\}=\max\{0,x_1\}$. Thus, we have a line at vertical line at $0$, and for $x_1<0$, then we have that this takes on the value of $x_1$ and for $x_1>0$, we have that this takes on the value of $0$, so this explains the figure with $x_1$ before the line and $0$ after the line (I assume this is the line at $0$). Thus, we get that the function will be linear and increasing with $x_1$ for $x_1<0$ and then constant for $x_1>0$. So if we were to "graph" this it would be a line that is increasing and then plateau's at $x_1=0$, and at this "corner" it is not linear. That is how I would interpret this, but I am not too sure what you mean. Either way, I hope this helps.