In an algebraic setting, our goal is to compare the tropical multiplicities introduced in 4.7 with the ones from Definition 7.5. Let us consider an algebraic variety $X$ over $K$ of dimension $n$ and an algebraic moment map $\varphi:X\rightarrow T={\mathbb{G}}_{m}^{r}$ over $K$. We assume that the map is generically finite. Note that ${\varphi_{\rm trop}}(X)={\rm Trop}(\overline{\varphi(X)})$. We conclude that ${\varphi_{\rm trop}}(X)$ is the support of an integral $\Gamma$-affine polyhedral complex of dimension $n$ endowed with the tropical multiplicities of ${\rm Trop}(\overline{\varphi(X)})$. Note that this tropical multiplicities are compatible with refinement and hence they define an integer valued function $m_{\rm alg}$ on the regular points of ${\varphi_{\rm trop}}(X)$. This means that the function is defined and constant in the relative interior of every $n$-dimensional polyhedron $\sigma$ contained in ${\varphi_{\rm trop}}(X)$ and if $\sigma$ is from the above integral $\Gamma$-affine polyhedral complex, then $m_{\rm alg}(\omega)$ is equal to the tropical multiplicity of $\sigma$ in ${\rm Trop}(\overline{\varphi(X)})$ for every $\omega\in{\rm relint}(\sigma)$.

The analytification ${X^{\rm an}}$ is not compact (unless $d=0$), but as $\partial X=\emptyset$, we can define tropical multiplicities in the same analytic manner as in Definition 7.5. Again, this is compatible with refinement and hence leads to a tropical multiplicity function $m_{\rm an}$ on the regular points of ${\varphi_{\rm trop}}(X)$.