} \oplus \frac{ \Z^2}{ <(-1, -2)> },
\]
where the differentials are the direct sums of the projection maps. Then under the differential we have $e_1 \mapsto (0, 2)$ and $e_2 \mapsto (1, -1)$ and the image of the differential is a proper sublattice of rank $2$ of $C^{BM}_1(Y; \F_1^Y)$. In fact we have $H^{BM}_1(Y; \F_1) = \Z_2$.
\end{exam}
%\newtheorem*{corTrichcomptor}{Corollary~\ref{cor:compactTorsionFree}}
\begin{coro*}[{Corollary~\ref{cor:compactTorsionFree}}]
If $Y$ is a compact non-singular tropical toric variety and $X$ is a combinatorially ample non-singular tropical hypersurface in $Y$, then all integral tropical homology groups of $X$ are torsion free.
\end{coro*}
\begin{proof}
By Proposition~\ref{prop:torichomo}, if $Y$ is compact, all its integral tropical homology groups are torsion free. Then by Theorem~\ref{thm:torsionfree}, all the integral tropical homology groups of $X$ are torsion free.
\end{proof}
%\newtheorem*{corTrichcompquasitor}{Corollary~\ref{cor:torsionfreequasiproj}}
\begin{coro*}[{Corollary~\ref{cor:torsionfreequasiproj}}]
Let $Y$ be a non-singular tropical toric variety associated to a fan whose support is a convex cone and such that the complex toric variety $Y_\C$ is quasi-projective. Let $X$ be a combinatorially ample non-singular tropical hypersurface in $Y$ such that $(Y, X)$ is a cellular pair and every parent face of a compact face of $Y$ is compact. Then both the standard and Borel--Moore integral tropical homology groups of $X$ are torsion free.
\end{coro*}
\begin{proof}
Assume that the convex cone supporting the fan of $Y$ is full dimensional in $\R^{n+1}$. We will first show that the tropical toric variety $Y$ equipped with the polyhedral structure dual to the polyhedral structure on its defining fan is a regular CW-complex. Thus the cellular tropical chain complexes can compute the standard and Borel--Moore homology groups of $Y$. To prove this claim, consider $Y_\C$, the quasi-projective toric variety associated to a fan $\Sigma$. Let $D$ be any ample Cartier divisor on $Y_{\C}$ and consider the associated polyhedron $P$ (see for example~\cite[Chapter~3]{FultonToric}). The hypothesis on the support of $\Sigma$ implies that it is the normal fan of $P$ (\cite[Chapter~6]{Mustata}). Therefore, the polyhedron $P$ is combinatorially isomorphic to $Y$, the tropical toric variety associated to $\Sigma$. Since $P$ is a polyhedron, it is a cell-complex in the sense of~\cite[Chapter~4]{CurryThesis}, and one can use the cellular description to compute the standard homology groups of $Y$.
As in the proof of Proposition~\ref{prop:torichomo}, both standard and Borel--Moore tropical homology groups of $Y$ vanish if $p>q$. It follows again from Poincar\'e duality and universal coefficients theorem that both standard and Borel--Moore tropical homology groups of $Y$ are torsion free. The statement for $X$ follows again from Theorem~\ref{thm:torsionfree}. Now suppose that the convex cone supporting the fan $\Sigma$ is of codimension $s$ in $\R^{n+1}$. Then the tropical toric variety $Y$ is a product $\R^s \times Y'$ where $Y'$ is a tropical toric variety of dimension $n+1-s$ satisfying the assumptions above. The tropical toric variety $Y'$ is then combinatorially isomorphic to a polyhedron $P'$. By the K\"unneth formula for Borel--Moore tropical homology~\cite[Theorem~B]{gross} we have
\[
H^{BM}_q\left(Y; \F_p^{Y}\right) = \bigoplus_{\substack{ i+j = p \\k +l = q}} H^{BM}_k\left(\R^s; \F_{i}^{\R^s}\right) \otimes H^{BM}_l\left(Y'; \F_{j}^{Y'}\right).
\]
Therefore, the Borel--Moore tropical homology groups of $Y$ are all torsion free and thus so are the standard tropical homology groups. This completes the proof of Corollary~\ref{cor:torsionfreequasiproj}.
\end{proof}
\section{Betti numbers of tropical homology and Hodge numbers}\label{section:epoly}
The $k$-compactly supported cohomology group of a complex hypersurface $X_{\C} \subset (\C^*)^{n+1}$ carries a mixed Hodge structure, see~\cite{DanilovKhovansky}. The numbers $e_c^{p,\,q}(X_{\C})$ are defined to be
\[
e_c^{p,\,q}(X_{\C}):= \sum_k (-1)^k h^{p,\,q}\left(H_c^k (X_{\C})\right),
\]
where $h^{p,\,q}(H_c^k (X_{\C}))$ denote the Hodge--Deligne numbers of $X_{\C}$. The numbers $e_c^{p,\,q}\linebreak(X_{\C})$ are the coefficients of the $E$-polynomial of $X_{\C}$,
\[
E\left(X_{\C};u,v\right) := \sum_{p,\,q} e_c^{p,\,q}(X_{\C})u^pv^q.
\]
The $\chi_y$ genus of $X_{\C}$ is defined to be
\[
\chi_y(X_{\C}) = E(X_{\C};y,1) := \sum_{p,\,q} e_c^{p,\,q}(X_{\C})y^p.
\]
Theorem~\ref{thm:Epoly} relates the coefficients of the $\chi_y$ genus and the Euler characteristics of the chain complexes $C_{\bullet}^{BM}(X; \F_p)$. For the proof of the theorem we require the notion of torically non-degenerate complex hypersurfaces.
\begin{defi}\label{def:torNonDeg}
If $Y_\C$ is a complex toric variety, a hypersurface $X_{\C} \subset Y_\C$ is \emph{torically non-degenerate} if the intersection of $X_{\C}$ with any torus orbit of $Y_\C$ is non-singular and $X_{\C}$ intersects each torus orbit of $Y_\C$ transversally. If $Y_\C$ is the complex toric variety associated to the Newton polytope of $X_\C$, then the second condition follows from the first one (see for example~\cite{Kho77}).
\end{defi}
%\newtheorem*{thmTricheEpoly}{Theorem~\ref{thm:Epoly}}
\begin{theo*}[{Theorem~\ref{thm:Epoly}}]
Let $X$ be an $n$-dimensional non-singular tropical hypersurface in a non-singular tropical toric variety $Y$. Let $X_{\C}$ be a complex hypersurface torically non-degenerate in the complex toric variety $Y_{\C}$ such that $X$ and $X_\C$ have the same Newton polytope.
Then
\[
(-1)^p \chi \left(C^{BM}_{\bullet}\left(X; \F^X_p\right)\right) = \sum_{q = 0}^n e_c^{p,\,q}(X_{\C}),
\]
and thus
\[
\chi_y(X_{\C}) = \sum_{p = 0}^{n} (-1)^p \chi \left(C^{BM}_{\bullet}\left(X; \F^X_p\right)\right) y^p.
\]
\end{theo*}
\begin{proof}
Firstly, the variety $X_{\C}$ is stratified by its intersection with the open torus orbits of $Y_{\C}$. Moreover, the numbers $e_c^{p,\,q}(X_{\C})$ are additive along strata by~\cite[Proposition~1.6]{DanilovKhovansky}. So we have
\[
\sum_{q = 0}^n e_c^{p,\,q}(X_{\C}) = \sum_{\rho} \sum_{q = 0}^n e_c^{p,\,q}(X_{\C,\,\rho})
\]
for $X_{\C} = \sqcup_{\rho} X_{\C,\,\rho} $, where $X_{\C, \rho} := X_{\C} \cap Y_{\C,\,\rho}$ and $Y_{\C, \rho}$ is the open torus orbit corresponding to the face $\rho$ of the fan $\Sigma$ defining $Y$ and $Y_{\C}$.
The tropical hypersurface $X$ admits a stratification analogous to $X_{\C}$. The Euler characteristics of the chain complexes for cellular tropical Borel--Moore homology of $X$ satisfy the same additivity property. Namely,
\[
\chi\left(C^{BM}_{\bullet}\left(X; \F^X_p\right)\right) = \sum_{\rho} \chi\left(C^{BM}_{\bullet}\left(X_{\rho}; \F^{X_{\rho}}_p\right)\right).
\]
Moreover, for any face $\rho$ of the fan $\Sigma$ defining $Y$ and $Y_\C$, the Newton polytope of $X_{\C,\,\rho}$ is equal to the Newton polytope of $X_\rho$. In fact, since $X$ is proper in $Y$ and $X_{\C}$ intersects the boundary of $Y_{\C}$ properly, it is enough to prove it for $\rho$ a ray of $\Sigma$ and then proceed by recurrence. Up to a toric change of coordinates, one can assume that $\rho$ is a ray in direction $e_1=(1,0,\,\cdots,\,0)$. Then the hypersurface $X_{\C,\,\rho}$ is given by the polynomial $f^\C(0,x_2,\,\cdots,\,x_{n+1})$, where $f^\C$ is the polynomial defining $X_\C$. Similarly the tropical polynomial of $X_{\rho}$ is obtained from the tropical polynomial of $X$ by removing all monomials containing $x_1$. So, the fact that $X$ and $X_{\C}$ have the same Newton polytope implies that $X_{\C,\,\rho}$ and $X_\rho$ do as well. Therefore, it suffices to prove the statement for $X \subset \R^{n+1}$ and $X_{\C} \subset (\C^*)^{n+1}$.
We now assume that $X$ is in $\R^{n+1}$ and $X_{\C}$ is in $(\C^*)^{n+1}$. In~\cite[Section~5.2]{KatzStapledon}, Katz and Stapeldon give a formula for the $\chi_y$ genus of a torically non-degenerate hypersurface in the torus. Their formula utilizes regular subdivisions of polytopes to refine the formula in terms of Newton polytopes of Danilov and Khovanskii~\cite{DanilovKhovansky}. Note that they use the term sch\"on in exchange for torically non-degenerate. Let $\Delta$ be the Newton polytope for $X_{\C}$ and $\tilde{\Delta}$ a regular subdivision of the lattice polytope $\Delta$. Then the formula is {
\begin{equation}\label{eq:chiyC}
\chi_y(X_{\C}) = \sum_{\substack{F\,\subset\, \tilde{\Delta} \\ F\,\not \subset\,\partial \Delta}} (-1)^{n+1-\dim F}\chi_y(X_{\C,\,F}),
\end{equation}
} where $X_{\C,\,F}$ is the hypersurface in the torus $(\C^*)^{n+1}$ defined by the polynomial obtained by restricting the polynomial defining $X_{\C}$ to the monomials corresponding to the lattice points in the face $F$ of $\tilde{\Delta}$. Notice our description of $X_{\C,\,F}$ differs from the one in~\cite{KatzStapledon} up to the direct product with a torus.
\goodbreak
Suppose that $\tilde{\Delta}$ is a primitive regular subdivision of $\Delta$. Then for each face $F$ of $\tilde{\Delta}$ the variety $X_{\C,\,F}$ is the complement of a hyperplane arrangement. By~\cite{Shapiro} its mixed Hodge structure is pure and
\[
\chi_y(X_{\C,\,F}) = \sum_{p = 0}^n (-1)^{n+p} \dim H^{n+p}_c(X_{\C,\,F})y^p.
\]
In fact, this hyperplane arrangement complement is $\CC_{n-q} \times (\C^*)^{q}$, where $\dim F\linebreak= n + 1- q $ and $\CC_{n-q}$ is the complement of $n+2-q$ generic hyperplanes in $\C P^{n-q}$. By~\cite{Zharkov13}, we have $\dim H^p(X_{\C,\,F}) = \rank \F_p(\sigma_F)$ where $\sigma_F$ is the face of the tropical hypersurface $X$ dual to $F$. By Poincar\'e duality for $X_{\C, \,F}$ we obtain $\dim H^p_c(X_{\C,\,F}) = \rank \F_{n-p}(\sigma_F)$.
Therefore, we obtain the formula
\[
\chi_y(X_{\C,\,F}) = y^{-1}(y - 1) ^q \left[(y-1)^{n+1-q} - (-1)^{n+1-q}\right].
\]
Therefore when the subdivision is primitive $\chi_y(X_{\C,\,F})$ only depends on the dimension of $F$. Moreover, if $\tilde{\Delta}$ is the subdivision dual to the tropical hypersurface $X$ then formula in Equation~\eqref{eq:chiyC} can be expressed in terms of the $f$-vector of bounded faces of $X$. Namely,
\begin{equation}\label{eqn:chiyX}
\chi_y(X_{\C}) = \sum_{q = 0}^{n} (-1)^qy^{-1}(y - 1) ^q \left[(y-1)^{n+1-q} - (-1)^{n+1-q}\right] f^b_q,
\end{equation}
where $f^b_q$ denotes the number of bounded faces of $X$ of dimension $q$.
On the other hand we can compute the Euler characteristics of the Borel--Moore chain complexes
\begin{equation}\label{eqn:EulerBM}
\chi (C^{BM}_{\bullet}(X; \F_p)) = \sum_{\tau\,\in \,X} (-1)^{\dim \tau} \rank \F_p(\tau).
\end{equation}
The star of a face $\tau$ of $X$ is a basic open subset and satisfies Poincar\'e duality from~\cite{JRS}. Therefore, we have
\begin{align*}
\rank \F_p(\tau) &= \rank H_0(\STar(\tau); \F_{p}) = \rank H^n_c(\STar(\tau) ; \F^{n-p}) \\
&= \sum_{\sigma\,\supset\,s\tau \dim\sigma=q} (-1)^{n-q} \rank \F_{n-p}(\sigma).
\end{align*}
since $\rank \F^{n-p}(\tau) = \rank \F_{n-p}(\tau)$ and also $ H^n_c(\STar(\tau) ; \F^{n-p}) $ is torsion free. Swapping the order of the sum we obtain
\[
\chi \left(C^{BM}_{\bullet}(X; \F_p)\right) = \sum_{\sigma\,\in\,X} (-1)^{n-\dim \sigma} \rank \F_{n-p}(\sigma) \sum_{\tau\,\subset\,\sigma} (-1)^{\dim \tau}.
\]
If $\sigma$ is a bounded face of $X$, then $\sum_{\tau \subset \sigma} (-1)^{\dim \tau} = 1$. If $\sigma$ is an unbounded face of $X$ then $\sum_{\tau \subset \sigma} (-1)^{\dim \tau} = 0$, since the one point compactification of $\sigma$ has Euler characteristic equal to $1$. Therefore, the sum in Equation~\eqref{eqn:EulerBM} becomes
\[
\chi \left(C^{BM}_{\bullet}\left(X; \F_p\right)\right) = \sum_{\substack{\tau\,\in\,X \\ \tau \text{ bounded}}} (-1)^{n-\dim \tau} \rank \F_{n-p}(\tau).
\]
For a face $\tau$ of dimension $q$ we have
\begin{align*}
\sum_{p = 0}^n (-1)^p \rank \F_{n-p}(\tau)y^p & = (-1)^n y^n \chi_ \tau(\frac{1}{y}) \\
& = (-1)^n y^{-1} (y - 1) ^q \left[(y-1)^{n+1-q} - (-1)^{n+1-q}\right],
\end{align*}
where $ \chi_ \tau$ is the polynomial from Corollary~\ref{cor:EPpoly}. By comparing this with Equation~\eqref{eqn:chiyX} we obtain
\[
\chi_y(X_{\C}) = \sum_{p = 0}^{n} (-1)^p \chi \left(C^{BM}_{\bullet}\left(X; \F^X_p\right)\right) y^p,
\]
and the proof of the Theorem~\ref{thm:Epoly} is complete.
\end{proof}
%\newtheorem*{corHodgeZ}{Corollary~\ref{cor:hodgeZtrop}}
\begin{coro*}[{Corollary~\ref{cor:hodgeZtrop}}]
Let $X$ be a non-singular and combinatorially ample compact tropical hypersurface in a non-singular compact tropical toric variety $Y$ and assume that $X$ has Newton polytope $\Delta$. Let $X_\C$ be a torically non-degenerate complex hypersurface in the compact complex toric variety $Y_{\C}$ also with Newton polytope $\Delta$. Then for all $p$ and $q$ we have
\[
\dim H^{p,\,q}(X_\C) = \rank H_q\left(X; \F^X_p\right).
\]
\end{coro*}
\begin{proof}
By combining Proposition~\ref{prop:torichomo} with the Lefschetz hyperplane section theorems for tropical homology and the homology of complex hypersurfaces of toric varieties, for $p + q < n$, we have
\begin{equation}\label{eq:hodgebetti}
\rank H_{q}(X; \F_p) = \rank H_{q}\left(Y; \F^Y_p\right) = h^{p,\,q}(Y_{\C}) = h^{p,\,q}(X_{\C}).
\end{equation}
The above equations combined with the Poincar\'e duality statements for all of $X, Y, X_{\C}$ and $Y_{\C}$ establishes the same equalities when $p + q > n$.
Therefore, it only remains to prove the statement when $q = n-p$. It follows from the tropical and complex versions of Lefschetz theorems and from Proposition~\ref{prop:torichomo} that
\begin{align*}
\chi\left(C^{BM}_{\bullet}\left(X; \F^X_p\right)\right) &= (-1)^p \rank H_{p}\left(Y; \F^Y_p\right) + (-1)^{n-p} \rank H_{n-p}\left(X; \F^X_p\right),
\\
\intertext{and}
\sum_q e_c^{p,\,q}(X_{\C}) &= \dim H^{p,\,p}(Y_{\C}) + (-1)^{n} \dim H^{p,\,n-p}(X_{\C})
\end{align*}
for $p\neq \tfrac{n}{2}$.
For $p=\tfrac{n}{2}$, we get
\begin{align*}
\chi\left(C^{BM}_{\bullet}\left(X; \F^X_\frac{n}{2}\right)\right) &= (-1)^{\frac{n}{2}} \rank H_{\frac{n}{2}}\left(X; \F^X_\frac{n}{2}\right),
\\
\intertext{and}
\sum_q e_c^{\frac{n}{2},\,q}(X_{\C}) &= \dim H^{\frac{n}{2},\,\frac{n}{2}}(X_{\C}).
\end{align*}
Again by Proposition~\ref{prop:torichomo} for tropical toric varieties we have
\[
\rank H_{p}\left(Y; \F^Y_p\right) = \dim H^{p,\,p}(Y_{\C}).
\]
The statement of the corollary follows after applying Theorem~\ref{thm:Epoly}.
\end{proof}
The next corollary also follows from Theorem~\ref{thm:Epoly}.
\begin{coro}\label{cor:affine}
Let $Y$ be a non-singular tropical toric variety associated to a fan whose support is a convex cone of maximal dimension in $\R^{n+1}$ and such that the complex toric variety $Y_\C$ is affine. Let $X$ be a combinatorially ample non-singular tropical hypersurface in $Y$ such that $(Y, X)$ is a cellular pair and every parent face of a compact face of $Y$ is compact. If $X_\C$ is a torically non-degenerate complex hypersurface in $Y_\C$ with the same Newton polytope as $X$, then
\[
\rank H_{q}^{BM}(X;\F_p)=
\begin{cases}
\displaystyle{\sum_{l=0}^{q}} h^{p,\,l}\left(H^{n}_c(X_\C)\right) & \text{ if } p+q=n \\
h^{p,\,p}\left(H^{2p}(X_\C)\right) & \text{ if } p=q>\frac{n}{2} \\
0 & \text{ otherwise}.
\end{cases}
\]
\end{coro}
\begin{proof}[Proof of Corollary~\ref{cor:affine}]
It follows from~\cite[Theorem~3.6]{CMM} that if $p\neq q$ or $k\neq 2p$, then $h^{p,\,q}(H^k_c(Y_\C))=0$. Therefore, if $p\neq q$, then $e^{p,\,q}(Y_\C)=0$ and when $p = q$ we have
\[
e_c^{p,\,p}(Y_\C)=h^{p,\,p}\left(H^{2p}_c(Y_\C)\right).
\]
From the proof of Corollary~\ref{cor:torsionfreequasiproj}, we also have $H^{BM}_q(Y; \F_p)= 0$ if $p\neq q$. The equality in Theorem~\ref{thm:Epoly} also holds if we replace $X$ and $X_{\C}$ with non-singular toric varieties $Y$ and $Y_{\C}$. This is because it holds for $(\C^*)^{k}$ and the Euler characteristic of the Borel--Moore complexes and the $\chi_y$ genus are both additive. Therefore, we obtain
\[
\rank H^{BM}_{p}\left(Y; \F^Y_p\right) = h^{p,\,p}\left(H^{2p}_c(Y_\C)\right).
\]
Notice that since $Y_\C$ is affine, the Andreotti--Frankel theorem imply that $h^{p,\,p}(H^{2p}_c\linebreak(Y_\C))=0$ if $2p