We define tropical rational function semifields $\overline{ \boldsymbol{ T}(X_1, \ldots, X_n)}$ and prove that a tropical curve $\varGamma$ is realized (except for points at infinity) as the congruence variety $V \subset \boldsymbol{ R}^n$ associated with a congruence on $\overline{ \boldsymbol{ T}(X_1, \ldots, X_n)}$ by giving a specific map $\varGamma \to V$. Also, we shed light on the relation between congruences $E$ on $\overline{ \boldsymbol{ T}(X_1, \ldots, X_n)}$ and congruence varieties associated with them and reveal the quotient semifield $\overline{ \boldsymbol{ T}(X_1, \ldots, X_n)} / E$ to play the role of coordinate rings that determine isomorphism classes of affine varieties in the classical algebraic geometry.
Categories: math.AG