Riemann-Roch inequality for smooth tropical toric surfaces

For a divisor $D$ on a tropical variety $X$, we define two amounts in order to estimate the value of $h^0(X,D)$, which are described by terms of global sections and computed more easily than $h^0(X,D)$. As an application of its estimation, we show that a Riemann-Roch inequality holds for smooth tropical toric surfaces.