Tropical geometry and integrable cellular automata I
Tropical spectral curve and isolevel set
We propose a method to study the integrable cellular automata via the tropical algebraic geometry. First we review the theory of tropical curves and Jacobi varieties introduced by Mikhalkin and Zharkov, and apply it to the spectral curve and the isolevel set of the ultra-discrete Toda lattice (UD-Toda) with periodic boundary condition. Next we introduce the interesting close relation between the periodic UD-Toda and the box and ball system.