The family $H^{\operatorname{af{}f}}_{s}$ can be thought of as an affine algebraic hypersurface in $(K^{*})^{d}$ defined over a complete algebraically closed field $K$ that contains $\mathbb{C}((s))$. Then, as the name suggests, the image of $H_{s}^{\operatorname{af{}f}}$ under the valuation map $val:(K^{*})^{d}\to R^{d}$ will be the non-Archimedean amoeba $\mathcal{A}^{\lambda}_{\infty}$ (cf. [Kap00]).

Interestingly, the potential functions for our Kähler forms $\omega_{s}$ will come from smoothing a convex piecewise linear function on $\mathbb{R}^{d}$. This gives another evidence that there may be a reformulation of mirror symmetry purely in non-Archimedean terms. There is a partial understanding of this approach [Kon00], but the entire program still remains wide open.