Tropical variety

Remarks

Fundamental theorem: let $K$ be algebraically closed with $\operatorname{val}$ nontrivial. Then $\operatorname{Trop}(I)$ is the closure in $\R^n$ of the points $(\operatorname{val}(z_1), \dots, \operatorname{val}(z_n))$, for $z$ a zero of $I$ in the torus $(K^\times)^n$. A typical such field is the field of Puiseux series over $\C$, with the valuation given by the lowest exponent. If moreover $I$ is prime and its zero set has dimension $d$, then $\operatorname{Trop}(I)$ is the support of a pure $d$-dimensional polyhedral complex.

Example: let $f = x + y + 1$ with all coefficients of valuation $0$. Then $\operatorname{trop}(f)(w_1, w_2) = \min(w_1, w_2, 0)$. The minimum is attained twice exactly on the three rays $\set{w_1 = w_2 \le 0}$, $\set{w_1 = 0 \le w_2}$ and $\set{w_2 = 0 \le w_1}$, which form the tropical line. Non-example: the point $(1, 1)$ is not on it, since there $\min(1, 1, 0) = 0$ is attained only by the constant term.

The intersection must run over all of $I$: the intersection of $\operatorname{Trop}(f)$ over a set of generators of $I$ can be strictly larger than $\operatorname{Trop}(I)$. A finite generating set for which equality holds is a tropical basis.