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This paper presents a study of the discrete Toda equation$(\tau_n^t)^2+\tau_{n-1}^t\tau_{n+1}^t=\tau_n^{t-1}\tau_n^{t+1}$, that wasintroduced in 1977. In this paper, it has been proved that the algebraicsolution of the discrete Toda equation, obtained via the Lax formalism, isnaturally related to the dual Grothendieck polynomial, which is a $K$-theoreticgeneralization of the Schur polynomial. A tropical permanent solution to theultradiscrete Toda equation has also been derived. The proposed method gives atropical algebraic representation of static solitons. Lastly, a new cellularautomaton realization of the ultradiscrete Toda equation has been proposed.

The discrete Toda equation revisited, dual $\beta$-Grothendieck polynomial, ultradiscretization and static soliton