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We give the addition formula for the tropical Hesse pencil, which is thetropicalization of the Hesse pencil parametrized by the level-three thetafunctions, via those for the ultradiscrete theta functions. The ultradiscretetheta functions are reduced from the level-three theta functions through theprocedure of ultradiscretization by choosing their parameters appropriately.The parametrization of the level-three theta functions firstly introduced in\cite{KKNT09} gives an explicit correspondence between the amoeba of the realpart of the Hesse cubic curve and the tropical Hesse curve. Moreover, throughthe parametrization, we obtain the subtraction-free forms of the additionformulae for the level-three theta functions, which lead to the additionformula for the tropical Hesse pencil in terms of the ultradiscretization.Using the parametrization, the tropical counterpart of the Hesse configurationis also given.

On the addition formula for the tropical Hesse pencil