$\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}=\sum_{i=1}^{n}I^{i}_{n}-\sum_{i=1}^{n}K^{i}_{n}\equiv
0\mod(2)$. Furthermore, since $\Delta^{i}_{n}\equiv 0\mod(d)$ and $d$ is odd,
then $\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}\equiv
0\mod(2d)$ and

$z_{n}^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}}=\left(D^{\frac{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}}{2d}}\right)^{2}\in(L^{\times})^{2}.$

By the induction hypothesis

$\displaystyle\operatorname{Tr}_{L/k}\left\langle\frac{\det\Delta^{i}}{d}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}\right\rangle=\sum\left\langle\alpha_{v}\right\rangle\text{
in }\operatorname{W}(k),$

where the sum runs over all $v\in\Lambda_{L}$ that are odd. Since every odd
corner vertex of $P$ belongs to $\Lambda_{L}$, and for every
$l\in\\{1,\dots,q\\}$ the coefficient
$\alpha_{v_{l}}=\epsilon_{P}(v_{l})\cdot\alpha_{l}$, our statement follows.

Case 2: $d$ is even. This implies that $\Delta^{i}_{n}\equiv 0\mod(2)$, for
all $i\in\\{1,\dots,n\\}$, and that
$\sum_{i=1}^{n}K^{i}_{n}=\sum_{i=1}^{n}I^{i}_{n}-\delta_{{K}^{i},{J}^{i}}\Delta^{i}_{n}\equiv\sum_{i=1}^{n}I^{i}_{n}\mod(2)$.

If $\sum_{i=1}^{n}I^{i}_{n}\equiv 0\mod(2)$, then

$\displaystyle\operatorname{Tr}_{E/L}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}z_{\circ}^{\sum_{i=1}^{n}I^{i}-e}\right\rangle$
$\displaystyle=\operatorname{Tr}_{E/L}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}z_{n}^{\sum_{i=1}^{n}I_{n}^{i}-1}\right\rangle$
$\displaystyle=\operatorname{Tr}_{E/L}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}\cdot
z_{n}\right\rangle$
$\displaystyle\overset{\ref{prop:traces}}{=}\frac{d}{2}h\in\operatorname{GW}(k),$

which implies that

$\operatorname{Tr}_{E/k}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}z_{\circ}^{\sum_{i=1}^{n}I^{i}-e}\right\rangle=0\in\operatorname{W}(k),$

by Proposition 2.13. On the other hand, there are not odd vertices of $P$
since for every vertex the sum
$\sum_{i=1}^{n}K^{i}_{n}\equiv\sum_{i=1}^{n}I^{i}_{n}\equiv 0\mod(2)$ and our
statement follows.

Alternatively, if $\sum_{i=1}^{n}I^{i}_{n}\equiv 1\mod(2)$, then

$\displaystyle\operatorname{Tr}_{E/L}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}z_{\circ}^{\sum_{i=1}^{n}I^{i}-e}\right\rangle$
$\displaystyle=\operatorname{Tr}_{E/L}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}z_{n}^{\sum_{i=1}^{n}I_{n}^{i}-e_{n}}\right\rangle$
$\displaystyle=\operatorname{Tr}_{E/L}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}\right\rangle$
$\displaystyle\overset{\ref{prop:traces}}{=}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}\right\rangle+\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}\hat{z}_{\circ}^{\sum_{i=1}^{n}\hat{I}^{i}-\hat{e}}D\right\rangle,$

in $\operatorname{W}(k)$. Let $P_{L_{0}}$ and $P_{L_{1}}$ be the convex hull
of the sets

$\Lambda_{L_{0}}=\left\\{\sum_{i=1}^{n}{K}^{i}_{n}\middle|K^{i}=I^{i}\text{ or
}J^{i},\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\left(\frac{\Delta^{i}_{n}}{d}\right)\equiv
0\mod(2)\right\\}\text{ and }$
$\Lambda_{L_{1}}=\left\\{\sum_{i=1}^{n}{K}^{i}_{n}\middle|K^{i}=I^{i}\text{ or
}J^{i},\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\left(\frac{\Delta^{i}_{n}}{d}\right)\equiv
1\mod(2)\right\\},$

respectively. Figure 10 (b) shows an example of a parallelepiped $P$ and
$P_{L_{0}}$ (the shaded parallelogram on the bottom) and $P_{L_{1}}$ (the
shaded parallelogram on top). Each set of corners $\Lambda_{L_{i}}$ have
$2^{n-1}$ elements and, analogously, each parallelepided $P_{L_{i}}$ can be
seen as a transversal cut of $P$, whose volume is $m$, hence $P_{L_{i}}$ is
not degenerated. To use our induction hypothesis (34) we have to show that
each set $\Lambda_{L_{i}}$ along with the coefficients $\alpha_{v}$ defined as
in (35) satisfy the equation (33). There exists an index
$i_{0}\in\\{1,\dots,n\\}$ such that
$\left(\frac{\Delta^{i_{0}}_{n}}{d}\right)\equiv 1\mod(2)$. Without loss of
generality, assume that $i_{0}=n$. Put
$v_{\circ}=\sum_{i=1}^{n}{I}^{i}\in\Lambda_{L_{0}}$ and
$w_{\circ}=\sum_{i=1}^{n-1}\hat{I}^{i}+\hat{J}^{n}\in\Lambda_{L_{1}}$. An
analogous computation as in the precedent case yields

$\alpha_{v}\hat{z}_{\circ}^{\hat{v}-\hat{e}}=\alpha_{v_{\circ}}\cdot\hat{z}_{\circ}^{\hat{v}_{\circ}-\hat{e}}\cdot
z_{n}^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}}=\alpha_{w_{\circ}}\cdot\hat{z}_{\circ}^{\hat{w}_{\circ}-\hat{e}}\cdot
z_{n}^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}-\Delta^{n}_{n}}.$

If $v\in\Lambda_{L_{0}}$, then

$z_{n}^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}}=D^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\frac{\Delta^{i}_{n}}{d}}\in(L^{\times})^{2}.$

If $v\in\Lambda_{L_{1}}$, then

$z_{n}^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\Delta^{i}_{n}-\Delta^{n}_{n}}=D^{\sum_{i=1}^{n}\delta_{{K}^{i},{J}^{i}}\cdot\frac{\Delta^{i}_{n}}{d}-\frac{\Delta^{n}_{n}}{d}}\in(L^{\times})^{2}.$

Furthermore, we have that

$\alpha_{v_{\circ}}\cdot\hat{z}_{\circ}^{\hat{v}_{\circ}-\hat{e}}\cdot
D=\alpha_{w_{\circ}}\cdot\hat{z}_{\circ}^{\hat{w}_{\circ}-\hat{e}}\cdot
z_{n}^{-\Delta^{n}_{n}}\cdot
D=\alpha_{w_{\circ}}\cdot\hat{z}_{\circ}^{\hat{w}_{\circ}-\hat{e}}\cdot
D^{-\frac{\Delta^{n}_{n}}{d}+1}=\alpha_{w_{\circ}}\cdot\hat{z}_{\circ}^{\hat{w}_{\circ}-\hat{e}}\in
L^{\times}/(L^{\times})^{2}.$

Therefore,

$\displaystyle\operatorname{Tr}_{E/k}\left\langle\det\Delta^{i}\prod_{i=1}^{n}\alpha_{i}z_{\circ}^{\sum_{i=1}^{n}I^{i}-e}\right\rangle$
$\displaystyle=\operatorname{Tr}_{L/k}\left(\left\langle\det\Delta^{i}\alpha_{v_{\circ}}\hat{z}_{\circ}^{\hat{v}_{\circ}-\hat{e}}\right\rangle+\left\langle\det\Delta^{i}\alpha_{\hat{w}_{\circ}}\hat{z}_{\circ}^{w_{\circ}-\hat{e}}\right\rangle\right)$
$\displaystyle=\sum\left\langle\alpha_{v}\right\rangle\in\operatorname{W}(k)$

where the sum runs over all lattice points $v\in\Lambda_{L_{0}}$ and
$v\in\Lambda_{L_{1}}$ such that are odd. Those are exactly the odd vertices in
$P$ since every vertex in $P$ has an odd $n$-th entry in this case.

∎

##  6 Enriched Tropical Bézout and Bernstein-Kushnirenko theorems

In sections 4 and 5 we assumed our algebras $E$ to be finite étale over the
field $k$ in order to take the trace. Since we need this assumption for any
point of intersection, we assume from now on that $k$ is a perfect field of
characteristic not equal to $2$. We use properties of toric varieties to give
applications of the computation we obtained in the precedent sections. For
more details on toric varieties we refer to [Ful16] and [GKZ94].

###  6.1 A tropical proof of Bézout’s theorem enriched in
$\operatorname{GW}(k)$

With the combinatorial formulas in Theorem 4.7 and in Theorem 5.6 for the
enriched intersection multiplicity, we can quadratically enrich Bernd
Sturmfels’ proof of the tropical Bézout theorem. The resulting count agrees
with Stephen McKean’s nontropical Bézout’s theorem 2.23 in the relatively
orientable case. In the non-relatively orientable case, we do not get an
invariant result for the sum of enriched intersection multiplicities at the
intersection points. However, our methods tell us all possible counts for this
sum.

The proof of the enriched tropical Bézout theorem for curves is an easy
corollary of the following Proposition.

######  Proposition 6.1.

Let $C_{1}$ and $C_{2}$ be tropical curves with Newton polygons $\Delta_{1}$
and $\Delta_{2}$, respectively. Let $v$ be a lattice point in the interior of
$\Delta_{1}+\Delta_{2}$. Let

$P_{v}\coloneqq\\{\text{$P$ in $\operatorname{DS}(C_{1}\cup C_{2})$ dual to
some $p\in C_{1}\cap C_{2}$, s.t. $v$ is a corner vertex of $P$}.\\}$

If the curves intersect tropically transversally, then

  1. 1.

The cardinality $\operatorname{Card}(P_{v})$ is even.

  2. 2.

There are equally many parallelograms $P$ in $P_{v}$ such that the sign
$\epsilon(v)$ (as defined in (15)) in $P$ is positive as there are with
negative sign

$\operatorname{Card}(\\{P\in
P_{v}\mid\epsilon_{P}(v)=+1\\})=\operatorname{Card}(\\{P\in
P_{v}\mid\epsilon_{P}(v)=-1\\}).$

###### Proof.

Assume there are $N$ edges in the dual subdivision
$\operatorname{DS}(C_{1}\cup C_{2})$ with vertex $v$. Each of these edges
either corresponds to an edge of $C_{1}$ or an edge of $C_{2}$. We assign an
element $i\in\mathbb{Z}/N\mathbb{Z}$ to each edge by walking around $v$
anticlockwise starting at some arbitrary edges with vertex $v$. Then the edges
$i$ and $i+1$ correspond to different curves if and only if they are edges of
a parallelogram corresponding to an intersection of $C_{1}$ and $C_{2}$.
Since, $N+1=1$ in $\mathbb{Z}/N\mathbb{Z}$, there needs to be an even number
of $i\in\mathbb{Z}/N\mathbb{Z}$ such that $i$ and $i+1$ correspond to
different curves. Hence, there is an even number of parallelograms with vertex
$v$ which correspond to an intersection of $C_{1}$ and $C_{2}$. Furthermore,
the number of $i$ where $i$ corresponds to an edge in $C_{1}$ and $i+1$
corresponds to an edge in $C_{2}$ must be equal to the number of $i$ where $i$
corresponds to an edge in $C_{2}$ and $i+1$ corresponds to an edge in $C_{1}$.
Now note that the first case corresponds to an intersection with positive sign
and the latter to an intersection with negative sign (see (15)). ∎

######  Example 6.2.

In Figure 8 there are three odd points in the interior of the Newton polygon.
The points $(3,1)$ and $(1,3)$ are both vertices of two parallelograms
corresponding to intersection points, while $(1,1)$ is not a vertex of a
parallelogram. Summing up the intersection multiplicities found in Example 4.8
we get $3h$ which coincides with Stephen McKean’s enriched (non-tropical)
Bézout Theorem 2.23.

Let

$\Delta_{d}=\operatorname{Conv}\\{(0,0),(d,0),(0,d)\\}\subset\mathbb{R}^{2}$

be the Newton polygon of a general degree $d$ polynomial in $2$ variables.
Note that if $C_{1}$ and $C_{2}$ are tropical curves with Newton polygons
$\Delta_{d_{1}}$ and $\Delta_{d_{2}}$ than $C_{1}\cup C_{2}$ has Newton
polygon $\Delta_{d_{1}+d_{2}}$.

######  Corollary 6.3 (Enriched tropical Bézout for curves).

Let $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ be two enriched tropical
curves with Newton polygons $\Delta_{d_{1}}$ and $\Delta_{d_{2}}$,
respectively, such that $d_{1}+d_{2}\equiv 1\mod 2$. Further, assume that
$\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ intersect tropically transversely
at every intersection point $p\in\widetilde{C}_{1}\cap\widetilde{C}_{2}$. Then

$\sum_{p\in\widetilde{C}_{1}\cap\widetilde{C}_{2}}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})=\frac{d_{1}\cdot
d_{2}}{2}h\in\operatorname{GW}(k).$

###### Proof.

For $v$ in the interior of $\Delta_{d_{1}+d_{2}}$ let $N(v)$ be the number of
parallelograms in the dual subdivision of
$\widetilde{C}_{1}\cup\widetilde{C}_{2}$ which correspond to an intersection
of $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$. Call the coefficient of $v$ in
$\operatorname{DS(\widetilde{C}_{1}\cup\widetilde{C}_{2})}$ $\alpha_{v}$. By
Theorem 4.7 and Proposition 6.1 we have

$\displaystyle\sum_{p}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})=\sum_{v}\left(\frac{N(v)}{2}\left\langle\alpha_{v}\right\rangle+\frac{N(v)}{2}\left\langle-\alpha_{v}\right\rangle\right)\text{
in }\operatorname{W}(k)$

where the first sum runs over the intersection points of $\widetilde{C}_{1}$
and $\widetilde{C}_{2}$ and the second sum runs over the odd vertices in the
interior of $\Delta_{d_{1}+d_{2}}$. Since $\left\langle
a\right\rangle+\left\langle-a\right\rangle=0$ in $\operatorname{W}(k)$, we get
that
$\sum_{v}\left(\frac{n(v)}{2}\left\langle\alpha_{v}\right\rangle+\frac{n(v)}{2}\left\langle-\alpha_{v}\right\rangle\right)=0$
in $\operatorname{W}(k)$ and thus
$\sum_{p}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})$
equals a multiple of $h$ in $\operatorname{GW}(k)$. By Example 2.5 the
classical tropical Bézout theorem tells us the rank of the sum
$\sum_{p}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})$,
which equals $\sum_{p}\operatorname{mult}_{p}(C_{1},C_{2})=d_{1}\cdot d_{2}$.
Recall that an element of $\operatorname{GW}(k)$ is determined by its rank and
its image in the Witt group $\operatorname{W}(k)$ (see Remark 2.9). Thus we
get

$\sum_{p}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})=\frac{d_{1}\cdot
d_{2}}{2}\cdot h\text{ in }\operatorname{GW}(k).$

∎

We can generalize the proof of Proposition 6.1 to higher dimensions.

######  Proposition 6.4.

Let $V_{1},\ldots,V_{n}$ be $n$ tropical hypersurfaces in $\mathbb{R}^{n}$
with associated Newton polytopes $\Delta_{1},\ldots,\Delta_{n}$, respectively.
Let $v$ be an odd lattice point in the interior of
$\Delta_{1}+\ldots+\Delta_{n}$ and let

$P_{v}\coloneqq\\{\text{$P$ in $\operatorname{DS}(V_{1}\cup\ldots\cup V_{n})$
dual to some $p\in V_{1}\cap\ldots\cap V_{n}$, s.t. $v$ is a corner vertex of
$P$}\\}.$

If the hypersurfaces $V_{1},\ldots,V_{n}$ intersect tropically transversely,
then

  1. 1.

The cardinality $\operatorname{Card}(P_{v})$ of $P_{v}$ is even.

  2. 2.

There are equally many parallelepipeds $P$ in $P_{v}$ such that the sign
$\epsilon_{P}(v)$ (as defined in (28)) in $P$ is positive as there are with
negative sign

$\operatorname{Card}(\\{P\in
P_{v}\mid\epsilon_{P}(v)=+1\\})=\operatorname{Card}(\\{P\in
P_{v}\mid\epsilon_{P}(v)=-1\\}).$

###### Proof.

Due to the transversality hypothesis, the hypersurfaces $V_{1},\ldots,V_{n-1}$
intersect along a tropical curve $C\subset\mathbb{R}^{n}$. This curve
intersects $V_{n}$ tropically transversely. Let us denote by
$R_{v}\subset\mathbb{R}^{n}$ the region where the monomial of exponent $v$ is
maximal in the product of defining equations of the hypersurfaces
$V_{1},\ldots,V_{n}$. Since $v$ is an inner lattice point of the dual
polytope, the region $R_{v}$ is a bounded polytope. If $P_{v}$ is empty, our
assertion follows. Otherwise, let $p\in C\cap V_{n}$ be an intersection point
such that its dual polytope $P\in P_{v}$. Let $\gamma$ be the connected
component of $C\cap\partial R_{v}$ containing the point $p$. We claim that
$\gamma$ is a piece-wise linear path. Namely, the set $\gamma$ is formed by
two segments of edges of $C$ containing intersection points with $V_{n}$,
together with bounded edges. If a vertex $w$ of $C$ is in $\gamma$, its
valency in $\gamma$ is $2$, corresponding to the edges in $C$ adjacent to the
region where the monomial corresponding to $R_{v}$ is maximal. Since $R_{v}$
is a bounded polytope, the curve $\gamma$ is compact, having an endpoint $q$
that is in $V_{n}$. Indeed, the point $q\neq p$ cannot be a vertex of $C$, or
its valency in $\gamma$ would be $1$, hence it is an inner point of an edge of
$C$. Since there are no changes in the monomials where the maximum is achieved
in the interior of an edge, this change is produced by the hypersurface
$V_{n}$. Therefore, these paths $\gamma$ establish a pairing between the
intersection points of $C\cap V_{n}$ adjacent to $R_{v}$.

We transfer the frame in $p$ through $\gamma$ to show that the polytopes
corresponding to the endpoints of $\gamma$ have opposite sign at the vertex
$V$. For that, let us start by recalling that the sign $\epsilon_{P}(v)$ is
the sign of the determinant $\left(\Delta^{i}\right)_{i=1}^{n}$, where every
$\Delta^{i}$ has been oriented in such a way that ${v+\Delta^{i}\in P}$. This
oriented vector is the normal vector of the facet of $V_{i}$ pointing outwards
to the region $R_{v}$. Let us define
$w_{p}\coloneqq\bigwedge_{i=1}^{n-1}\Delta^{i}$ (the vector of alternating
minors of the $n-1\times n$ matrix $\left(\pm\Delta^{i}\right)_{i=1}^{n-1}$).
The vector $w_{p}$ is a director vector of the edge of $C$ containing $p$,
albeit not a primitive one. Moreover, the sign
$\epsilon_{P}(V)=\operatorname{sign}(\det\left(\Delta^{i}\right)_{i=1}^{n})=\operatorname{sign}(w_{p}\cdot\Delta^{n})$
can be computed as the sign of the inner product of $w_{p}$ with the normal
vector of the facet of $V_{n}$ containing $p$, oriented outwards the region
$R_{v}$. We can define $w_{\gamma(t)}$ for every point of $\gamma$ that is an
inner point of an edge of $C$. If we oriented $\gamma$ as a path starting at
$p$, at every point $\gamma(t)$, the orientations of $w_{\gamma(t)}$ and
$\gamma$ would have the same relation (either coincide or differ), since the
relative position of the normal vectors of the facets of $V^{i}$ at
$\gamma(t)$ does not change in the boundary of $R_{v}$. This implies that
exactly one of the vectors $w_{p}$ at $p\in V_{n}$ or $w_{q}$ at $q\in V_{n}$
is oriented towards the region $R_{v}$ while the other one is not. Hence, the
endpoints of $\gamma$ have opposite signs. ∎

As a corollary we get an enriched tropical Bézout theorem. For this let

$\Delta_{d}=\operatorname{Conv}\\{(0,0,\ldots,0),(d,0,\ldots,0),(0,d,0,\ldots,0),\ldots,(0,\ldots,0,d)\\}\subset\mathbb{R}^{n}.$

######  Corollary 6.5 (Enriched tropical Bézout).

Let $\widetilde{V}_{1},\ldots,\widetilde{V}_{n}$ be enriched tropical
hypersurfaces in $\mathbb{R}^{n}$ with Newton polytopes
$\Delta_{d_{1}},\ldots,\Delta_{d_{n}}$ such that $\sum_{i=1}^{n}d_{i}\equiv
n+1\mod 2$ and assume that $\widetilde{V}_{1},\ldots,\widetilde{V}_{n}$
intersect tropically transversally at every intersection point. Then

$\sum_{p\in\widetilde{V}_{1}\cap\ldots\cap\widetilde{V}_{n}}\widetilde{\operatorname{mult}}_{p}(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})=\frac{d_{1}\cdots
d_{n}}{2}h\in\operatorname{GW}(k).$

###### Proof.

Note that in the relatively orientable case, that is when
$\sum_{i=1}^{n}d_{i}\equiv n+1\mod 2$ all odd points in the dual subdivision
of $\widetilde{V}_{1}\cup\ldots\cup\widetilde{V}_{n}$ lie in the interior of
$\Delta_{d_{1}+\ldots+d_{n}}=\Delta_{1}+\ldots+\Delta_{n}$ and none on the
boundary. Now the proof follows from Proposition 6.4 by the same arguments as
in the proof of Corollary 6.3. ∎

######  Remark 6.6.

This yields a new proof for Bézout’s theorem enriched in
$\operatorname{GW}(k)$. Let

$V\coloneqq\mathcal{O}_{\mathbb{P}^{n}_{k}}(d_{1})\oplus\ldots\oplus\mathcal{O}_{\mathbb{P}^{n}_{k}}(d_{n})\rightarrow\mathbb{P}_{k}^{n}$

and let $V_{k\\{\\{t\\}\\}}$ be its base change to the field $k\\{\\{t\\}\\}$
of Puiseux series. Recall that McKean showed the enriched Bézout theorem by
computing the $\mathbb{A}^{1}$-Euler number of $V$. Furthermore, recall that
the $\mathbb{A}^{1}$-Euler number of $V_{k\\{\\{t\\}\\}}$ equals the sum of
local indices at the zeros of a general section of $V_{k\\{\\{t\\}\\}}$. A
general section of $V_{k\\{\\{t\\}\\}}$ is defined by $n$ general polynomials
over $k\\{\\{t\\}\\}$ which give rise to enriched tropical hypersurfaces
$\widetilde{V}_{1},\ldots,\widetilde{V}_{n}$. We defined the enriched
intersection multiplicities of the corresponding enriched tropical
hypersurface to be this local index. Thus it follows directly from Corollary
6.5 that

$n^{\mathbb{A}^{1}}(V_{k\\{\\{t\\}\\}})=\sum_{p\in\widetilde{V}_{1}\cap\ldots\cap\widetilde{V}_{n}}\widetilde{\operatorname{mult}}_{p}(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})=\frac{d_{1}\cdots
d_{n}}{2}h\in\operatorname{GW}(k\\{\\{t\\}\\}).$

Euler classes commute with base change. Hence, the image of the Euler class
$n^{\mathbb{A}^{1}}(V)$ under the map
$\operatorname{GW}(k)\rightarrow\operatorname{GW}(V_{k\\{\\{t\\}\\}})$ induced
by $k\rightarrow k\\{\\{t\\}\\}$ is $n^{\mathbb{A}^{1}}(V_{k\\{\\{t\\}\\}})$.
Since this map is an isomorphism and it sends a generator $\left\langle
a\right\rangle$ of $\operatorname{GW}(k)$ to $\left\langle
a\right\rangle\in\operatorname{GW}(k\\{\\{t\\}\\})$, we conclude that

$n^{\mathbb{A}^{1}}(V)=\frac{d_{1}\cdots d_{n}}{2}h\in\operatorname{GW}(k).$

####  6.1.1 Non-relatively orientable case

In the non-relatively orientable case, that is when
$\sum_{i=1}^{n}d_{i}\not\equiv n+1\mod 2$, we do not get an invariant count.
This can also be seen in our proof for the enriched tropical Bézout theorem:
In case $\sum_{i=1}^{n}d_{i}\not\equiv n+1\mod 2$, not all odd points are in
the interior of $\Delta_{d_{1}+\ldots+d_{n}}$, but some are on the boundary.
For these points on the boundary, we cannot apply Proposition 6.1.

######  Example 6.7.

Figure 11 shows the intersection of two tropical conics and the dual
subdivision of the union of the conics. There are two odd points on the
boundary of the Newton polygon of the union. We enrich the two tropical conics
by assigning coefficients. Then the sum over the enriched intersection
multiplicities equals

$h+\left\langle\alpha_{(3,1)},-\alpha_{(1,3)}\right\rangle$

where $\alpha_{(3,1)}$ is the coefficient of the vertex $(3,1)$ and
$\alpha_{(1,3)}$ is the coefficient of the vertex $(1,3)$ in the dual
subdivision. Hence, the sum depends on the choice of coefficients of the
enriched tropical conics, but there is always a hyperbolic summand.

(a) Intersection

(b) Dual subdivision

Figure 11: Intersection of two tropical conics

Let $\widetilde{V}_{1},\ldots,\widetilde{V}_{n}$ be enriched tropical
hypersurfaces in $\mathbb{R}^{n}$ with Newton polytopes
$\Delta_{d_{1}},\ldots,\Delta_{d_{n}}$ such that $\sum d_{i}\not\equiv n+1\mod
2$. As suggested in the example, we can find a lower bound for the number of
hyperbolic summands in
$\sum_{p}\widetilde{\operatorname{mult}}_{p}(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})$.
For $n$ odd let $m\in\mathbb{Z}$ such that the sum $d_{1}+\ldots+d_{n}=2m$ and
for $n$ even let $m\in\mathbb{Z}$ such that the sum $d_{1}+\ldots+d_{n}=2m+1$.
Set

$N(m)\coloneqq\text{the number of odd points on }\Delta_{d_{1}+\ldots+d_{n}}.$

The following table computes $N(m)$ for $n\leq 6$.

$\begin{array}[]{cc}d_{1}=2m+1&1\\\ d_{1}+d_{2}=2m&m\\\
d_{1}+d_{2}+d_{3}=2m+1&\displaystyle\frac{m(m+1)}{2}\\\
d_{1}+\dots+d_{4}=2m&\displaystyle\frac{m(m+1)(m+1)}{3!}\\\
d_{1}+\dots+d_{5}=2m+1&\displaystyle\frac{m(m+1)(m-1)(m+2)}{4!}\\\
d_{1}+\dots+d_{6}=2m&\displaystyle\frac{m(m+1)(m-1)(m+2)(m-2)}{5!}\\\
\end{array}$

Since the only non-hyperbolic contribution to
$\sum_{p\in\widetilde{V}_{1}\cap\ldots\cap\widetilde{V}_{n}}\widetilde{\operatorname{mult}}_{p}(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})$
comes from the odd points on the boundary, we get the following Corollary.

######  Corollary 6.8 (Enriched tropical Bézout in the non-relatively
orientable case).

Let $\widetilde{V}_{1},\ldots,\widetilde{V}_{n}$ be enriched tropical
hypersurfaces in $\mathbb{R}^{n}$ with Newton polygons
$\Delta_{d_{1}},\ldots,\Delta_{d_{n}}$ such that
$\sum_{i=1}^{n}d_{i}\not\equiv n+1\mod 2$. Then

$\sum_{\text{intersections
}p}\widetilde{\operatorname{mult}}_{p}(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})=\frac{d_{1}\cdots
d_{n}-r}{2}h+\left\langle
a_{1},\ldots,a_{r}\right\rangle\in\operatorname{GW}(k)$

where $r$ has to be smaller or equal the number $N(m)$ of odd points on
$\partial\Delta_{d_{1}+\ldots+d_{n}}$.

######  Remark 6.9.

In the non-relatively orientable case McKean shows that one can orient the
vector bundle

$V=\mathcal{O}_{\mathbb{P}^{n}}(d_{1})\oplus\ldots\oplus\mathcal{O}_{\mathbb{P}^{n}}(d_{n})\rightarrow\mathbb{P}^{n}$

relative to a divisor at infinity and compute the $\mathbb{A}^{1}$-Euler
number of $V$ relative to this divisor in the sense of Larson-Vogt [LV19].
Corollary 6.8 gives us a lower bound for the number of hyperbolic forms in
this $\mathbb{A}^{1}$-Euler number.

The lower bound on the number of hyperbolic summands is in Corollary 6.8 is
not necessarily strict. For enriched tropical curves we find a better, strict
bound.

######  Corollary 6.10 (Enriched tropical Bézout for curves in the non-
relatively orientable case).

Let $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ be two enriched tropical
curves of degree $\Delta_{d_{1}}$ and $\Delta_{d_{2}}$, respectively, with
$d_{1}+d_{2}\equiv 0\mod 2$ that intersect tropically transversely. Let
$d\coloneqq\min\\{d_{1},d_{2}\\}$. Then

$\sum_{p\in\widetilde{C}_{1}\cap\widetilde{C}_{2}}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})=\frac{d_{1}\cdot
d_{2}-d}{2}h+\left\langle
a_{1},\ldots,a_{d}\right\rangle\in\operatorname{GW}(k)$

for some $a_{1},\ldots,a_{d}\in k^{\times}/(k^{\times})^{2}$.

###### Proof.

In case, $d_{1}+d_{2}\equiv 0\mod 2$, there are $\frac{d_{1}+d_{2}}{2}$ odd
points on the boundary of $\Delta_{d_{1}}+\Delta_{d_{2}}$, all lying on the
hypotenuse. To get a non-hyperbolic summand, one of the two edges adjacent to
an odd vertex on the hypotenuse has to belong $\widetilde{C}_{1}$ and the
other one has to belong to $\widetilde{C}_{2}$. This can happen at most
$d=\min(d_{1},d_{2})$ times since only $d_{1}$ segments on the hypotenuse of
$\Delta_{d_{1}}+\Delta_{d_{2}}$ correspond to edges of $\widetilde{C}_{1}$ and
$d_{2}$ correspond to edges of $\widetilde{C}_{2}$. ∎

###  6.2 Bernstein-Kushnirenko theorem

The results above do not restrict to hypersurfaces in $\mathbb{P}^{n}$. The
tools of tropical geometry can be applied to toric varieties, where the action
of the torus yields a combinatorial approach to their study.

######  Example 6.11.

Let $C_{1}$ and $C_{2}$ be two curves in $\mathbb{P}^{1}\times\mathbb{P}^{1}$
defined by $f_{1}$ and $f_{2}$ of bidegree $(d_{1},d_{2})$ and
$(e_{1},e_{2})$, respectively. Then $f_{1}$ and $f_{2}$ define a section of

$V\coloneqq\mathcal{O}(d_{1},d_{2})\oplus\mathcal{O}(e_{1},e_{2})\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$

where
$\mathcal{O}(a,b)=\pi_{1}^{*}\mathcal{O}_{\mathbb{P}^{1}}(a)\otimes\pi_{2}^{*}\mathcal{O}_{\mathbb{P}^{1}}(b)$
and $\pi_{i}:\mathbb{P}^{1}\times\mathbb{P}^{1}\rightarrow\mathbb{P}^{1}$ is
the $i$th projection for $i=1,2$. The vector bundle $V$ is relatively
orientable if and only if $\det
V\otimes\omega_{\mathbb{P}^{1}\times\mathbb{P}^{1}}=\mathcal{O}(d_{1}+e_{1}-2,d_{2}+e_{2}-2)$
is a square which is the case if and only if both $d_{1}+e_{1}$ and
$d_{2}+e_{2}$ are even. The Newton polygons $\operatorname{NP}(f_{1})$ and
$\operatorname{NP}(f_{2})$ are rectangles with corners
$(0,0),(d_{1},0),(0,d_{2}),(d_{1},d_{2})$, respectively with corners
$(0,0),(e_{1},0),(0,e_{2}),(e_{1},e_{2})$. The Minkowski sum
$\operatorname{NP}(f_{1})+\operatorname{NP}(f_{2})$ is the rectangle with
corners $(0,0),(d_{1}+e_{1},0),(0,d_{2}+e_{2}),(d_{1}+e_{1},d_{2}+e_{2})$.
This rectangle $\operatorname{NP}(f_{1})+\operatorname{NP}(f_{2})$ has no odd
vertices on the boundary if and only if both $d_{1}+e_{1}$ and $d_{2}+e_{2}$
are even, that is exactly when the vector bundle $V$ is relatively orientable.
Let $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ be two enriched tropical
curves with Newton polytopes equal to $\operatorname{NP}(f_{1})$ and
$\operatorname{NP}(f_{2})$ and assume that both $d_{1}+e_{1}$ and
$d_{2}+e_{2}$ are even. Then Proposition 6.1 implies that

$\displaystyle\sum_{p\in C_{1}\cap
C_{2}}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})=\frac{d_{1}e_{2}+d_{2}e_{1}}{2}h\in\operatorname{GW}(k).$

Equivalently, we get that the $\mathbb{A}^{1}$-Euler number of the vector
bundle $V$ equals

$\displaystyle n^{\mathbb{A}^{1}}(V)=\sum_{p\in C_{1}\cap
C_{2}}\operatorname{ind}_{p}(f_{1},f_{2})$
$\displaystyle=\frac{\operatorname{Area}(\operatorname{NP}(f_{1})+\operatorname{NP}(f_{2}))-\operatorname{Area}(\operatorname{NP}(f_{1}))-\operatorname{Area}(\operatorname{NP}(f_{2}))}{2}h$
$\displaystyle=\frac{d_{1}e_{2}+d_{2}e_{1}}{2}h\in\operatorname{GW}(k)$

which yields an enriched count of intersection points of two curves in
$\mathbb{P}^{1}\times\mathbb{P}^{1}$.

######  Example 6.12.

Let $C_{1}$ and $C_{2}$ be two curves in the Hirzebruch surface $\Sigma_{n}$
defined by $f_{1}$ and $f_{2}$ of bidegree $(a_{1},b_{1})$ and
$(a_{2},b_{2})$, respectively, where $a_{i}=C_{i}\cdot F,b_{i}=C_{i}\cdot
E+na_{i}$, for $F$ a generic fiber and $E$ the exceptional divisor of
$\Sigma_{n}$. Then $f_{1}$ and $f_{2}$ define a section of

$V\coloneqq\mathcal{O}(a_{1}E+b_{1}F)\oplus\mathcal{O}(a_{2}E+b_{2}F)\rightarrow\Sigma_{n}.$

The vector bundle $V$ is relatively orientable if and only if

$\det
V\otimes\omega_{\Sigma_{n}}=\mathcal{O}\left((a_{1}+a_{2}-2)E+(b_{1}+b_{2}-(n+2)F)\right)$

is a square, which is the case if and only if $a_{1}+a_{2}$ is even and
$b_{1}+b_{2}\equiv n\mod 2$. The Newton polygons $\operatorname{NP}(f_{1})$
and $\operatorname{NP}(f_{2})$ are trapezia with corners
$(0,0),(a_{1}n+b_{1},0),(b_{1},a_{1}),(0,a_{1})$ and
$(0,0),(a_{2}n+b_{2},0),(b_{2},a_{2}),(0,a_{2})$, respectively. The Minkowski
sum $\operatorname{NP}(f_{1})+\operatorname{NP}(f_{2})$ is the trapezium with
corners
$(0,0),((a_{1}+a_{2})n+b_{1}+b_{2},0),(b_{1}+b_{2},a_{1}+a_{2}),(0,a_{1}+a_{2})$.
This trapezium $\operatorname{NP}(f_{1})+\operatorname{NP}(f_{2})$ has no odd
vertices on the boundary if and only if $a_{1}+a_{2}$ is even and
$b_{1}+b_{2}\equiv n\mod 2$, that is exactly when $V$ is relatively
orientable. Let $\widetilde{C}_{1}$ and $\widetilde{C}_{2}$ be two tropical
curves with Newton polytopes equal to $\operatorname{NP}(f_{1})$ and
$\operatorname{NP}(f_{2})$ and assume that $a_{1}+a_{2}$ is even and
$b_{1}+b_{2}\equiv n\mod 2$. Then Proposition 6.1 implies that

$\displaystyle\sum_{p\in\widetilde{C}_{1}\cap\widetilde{C}_{2}}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2})=\frac{a_{1}a_{2}n+a_{1}b_{2}+a_{2}b_{1}}{2}h\in\operatorname{GW}(k).$

Equivalently, as in the previous example, this coincides with
$n^{\mathbb{A}^{1}}(V)$ and yields an enriched count of intersection points of
two curves in $\Sigma_{n}$.

The examples above as well as Bézout’s theorem are special cases of a
quadratic enrichment of the Bernstein-Kushnirenko theorem. We recall the
classical statement of this theorem. Let $A$ be a finite subset of
$\mathbb{Z}^{n}$ and let

$L_{A}\coloneqq\left\\{f\middle|f(x)=\sum_{I\in A}c_{I}x^{I}=\sum_{I\in
A}c_{I}x_{1}^{I_{1}}\ldots x_{n}^{I_{n}}\text{, }c_{I}\in k^{\times}\right\\}$

be the space of Laurent polynomials whose exponents are in $A$. Let
$\Delta_{A}$ be the convex hull of the points in $A$. The classical Bernstein-
Kushnirenko theorem says.

######  Theorem 6.13 (Bernstein-Kushnirenko theorem).

For $n$ finite subsets $A_{1},\ldots,A_{n}$ of $\mathbb{Z}^{n}$ and for a
generic system of equations

$f_{1}(x)=\ldots=f_{n}(x)=0$

where $f_{i}\in L_{A_{i}}$, the number of solutions in
$(\mathbb{C}\setminus\\{0\\})^{n}$ equals the mixed volume

$\operatorname{MVol}(\Delta_{A_{1}},\ldots,\Delta_{A_{n}}).$

Before we state the quadratically enriched version of this theorem, we define
the following condition that can be seen as the combinatorial analogue of
relative orientability.

######  Definition 6.14.

We say that the tuple $(A_{1},\ldots,A_{n})$ is _combinatorially oriented_ if
the Minkowski sum $\Delta_{A_{1}}+\ldots+\Delta_{A_{n}}$ has no odd points on
the boundary.

######  Example 6.15 (Bézout).

Let $A_{i}=\Delta_{d_{i}}\cap\mathbb{Z}^{n}$ for some positive integer
$d_{i}$, i.e., $\Delta_{A_{i}}=\Delta_{d_{i}}$, for $i=1,\ldots,n$. Then
$(A_{1},\ldots,A_{n})$ is combinatorially oriented if and if
$\Delta_{d_{1}}+\ldots+\Delta_{d_{n}}$ has no odd boundary points which is
exactly the case if $d_{1}+\ldots+d_{n}\equiv n+1\mod 2$, i.e., exactly when
the vector bundle

$\mathcal{O}_{\mathbb{P}^{n}}(d_{1})\oplus\ldots\oplus\mathcal{O}_{\mathbb{P}^{n}}(d_{n})\rightarrow\mathbb{P}^{n}$

is relatively orientable.

In all examples 6.15, 6.11 and 6.12 above the condition of being
combinatorially oriented coincides with the condition for the corresponding
vector bundle to be relatively orientable which motivates the following
conjecture.

######  Conjecture 6.16.

Let $X$ be a smooth toric variety of dimension $n$ and let
$f_{1},\ldots,f_{n}$ be regular functions on $X$ such that $f_{i}$ is a non-
trivial section of a line bundle $\mathcal{L}_{i}\rightarrow X$ such that the
system $f_{1}=\dots=f_{n}=0$ has a non-empty solution set formed of isolated
zeros. Furthermore, let $A_{i}\coloneqq\\{\text{exponents of
}f_{i}\\}\subset\mathbb{Z}^{n}$. Then $(A_{1},\ldots,A_{n})$ is
combinatorially oriented if and only if the vector bundle
$\mathcal{L}_{1}\oplus\ldots\oplus\mathcal{L}_{n}\rightarrow X$ is relatively
orientable.

Let us say that a variety has the _combinatorially orientability property_ if
it this conjecture holds for every sum of $\dim X$ line bundles satisfying the
hypothesis. We prove in the following theorem that the class of varieties that
satisfy this conjecture is closed under products. In particular, this property
holds on products of projective spaces and Hirzebruch surfaces.

######  Theorem 6.17.

If $X_{1}$ and $X_{2}$ are smooth toric varieties that satisfy the
combinatorially orientability property, then $X_{1}\times X_{2}$ also
satisfies the combinatorially orientability property.

###### Proof.

The product $X\coloneqq X_{1}\times X_{2}$ has a toric structure given by the
product of the toric structures on each component. Since $X$ is smooth, we
have that

$\operatorname{Pic}(X)\simeq H^{2}(X,\mathbb{Z})=H^{2}(X_{1},\mathbb{Z})\oplus
H^{2}(X_{2},\mathbb{Z}).$

by the Künneth formula and the fact that $H^{1}(X_{i},\mathbb{Z})=0$ due to
$X_{i}$ being smooth, $i=1,2$. Hence, through this isomorphism, every line
bundle $\mathcal{L}$ is determined by its bidegree $\bar{d}=(d_{1},d_{2})$,
where each degree class $d_{i}\in H^{2}(X_{i},\mathbb{Z}),i=1,2$. Therefore,
for every line bundle $\mathcal{L}$ over $X$ of degree $\bar{d}$, there are
line bundles $\mathcal{L}^{1}$ and $\mathcal{L}^{2}$ over $X_{1}$ and $X_{2}$
of degree $d_{1}$ and $d_{2}$, respectively, such that

$\mathcal{L}=p_{1}^{*}\mathcal{L}^{1}\otimes p_{2}^{*}\mathcal{L}^{2}$

where $p_{i}:X\longrightarrow X_{i},i=1,2$ is the component projection. Given
this decomposition, the line bundle $\mathcal{L}$ is a square if and only if
each $\mathcal{L}^{i}$, $i=1,2$, is a square. Moreover, the polytope $\Delta$
associated to $\mathcal{L}$ in $\Lambda\otimes\mathbb{R}$ is the product of
the polytopes $\Delta_{1}$ and $\Delta_{2}$ associated to $\mathcal{L}_{1}$
and $\mathcal{L}_{2}$ in $\Lambda_{1}\otimes\mathbb{R}$ and
$\Lambda_{2}\otimes\mathbb{R}$, where $\Lambda_{1}$ and $\Lambda_{2}$ are the
lattices associated to $X_{1}$ and $X_{2}$, respectively, and the lattice
$\Lambda\coloneqq\Lambda_{1}\times\Lambda_{2}$ is the one associated to $X$.
The polytope $\Delta$ has boundary

$\partial(\Delta)=\partial(\Delta^{1}\times\Delta^{2})=(\partial(\Delta^{1})\times\Delta^{2})\cup(\Delta^{1}\times\partial(\Delta^{2})),$

and so, the odd lattice points are
$\partial(\Delta)^{\mathrm{odd}}=(\partial(\Delta^{1})^{\mathrm{odd}}\times(\Delta^{2})^{\mathrm{odd}})\cup((\Delta^{1})^{\mathrm{odd}}\times\partial(\Delta^{2})^{\mathrm{odd}}).$
Now, let $V$ be the vector bundle
$\mathcal{L}_{1}\oplus\ldots\oplus\mathcal{L}_{n}\rightarrow X$, where $n=\dim
X$, $L_{i}$ is a line bundle over $X$ and $A_{i}$ is the exponent set of a
generic section $f_{i}$ of $\mathcal{L}_{i},i=1,\dots,n$. Put
$\Delta_{i}=\operatorname{Conv}(A_{i})$ the convex hull of the set $A_{i}$ in
$\Lambda\otimes\mathbb{R}.$ Assume that the system $f_{1}=f_{2}=\dots=f_{n}=0$
has an isolated zero. In this case we have that
$(\sum_{i=1}^{n}\Delta_{i})^{\mathrm{odd}}\neq\emptyset$, otherwise there
would be an $i_{0}$ for which $\Delta_{i_{0}}=\\{\mathrm{pt}\\}$ or there
would be a vector subspace $H\subset\Lambda\otimes\mathbb{R}$ of lower
dimension, containing all $\Delta_{i}\subset H$, which contradicts the fact
that the system has only isolated zeros. This implies that
$(\sum_{i=1}^{n}\Delta_{i}^{1})^{\mathrm{odd}}\neq\emptyset$ and
$(\sum_{i=1}^{n}\Delta_{i}^{2})^{\mathrm{odd}}\neq\emptyset$. Lastly, since
the Minkowski sum commutes with products, the odd lattice points in the
boundary of the Minkowski sum satisfy

$\partial(\sum_{i=1}^{n}\Delta_{i})^{\mathrm{odd}}=(\partial(\sum_{i=1}^{n}\Delta_{i}^{1})^{\mathrm{odd}}\times(\sum_{i=1}^{n}\Delta_{i}^{2})^{\mathrm{odd}})\cup((\sum_{i=1}^{n}\Delta_{i=1}^{1})^{\mathrm{odd}}\times\partial(\sum_{i=1}^{n}\Delta_{i}^{2})^{\mathrm{odd}}).$

These facts imply our statement, namely if the $n$-tuple
$(A_{1},\ldots,A_{n})$ is combinatorially oriented, then
$\partial(\sum_{i=1}^{n}\Delta_{i})^{\mathrm{odd}}=\emptyset$ by definition.
Since $(\sum_{i=1}^{n}\Delta_{i}^{1})^{\mathrm{odd}}\neq\emptyset$ and
$(\sum_{i=1}^{n}\Delta_{i}^{2})^{\mathrm{odd}}\neq\emptyset$ in this case, we
have that the $n$-tuple $(A_{1},\ldots,A_{n})$ is combinatorially oriented if
and only if both of the sets
$\partial(\sum_{i=1}^{n}\Delta_{i}^{1})^{\mathrm{odd}}$ and
$\partial(\sum_{i=1}^{n}\Delta_{i}^{2})^{\mathrm{odd}}$ are empty. Since
$X_{1}$ and $X_{2}$ satisfy the combinatorially orientability property, the
sets $(\partial(\sum_{i=1}^{n}\Delta_{i}^{1})^{\mathrm{odd}}$ and
$(\partial(\sum_{i=1}^{n}\Delta_{i}^{2})^{\mathrm{odd}}$ are empty if and only
if the vector bundles given by the direct sum of the components of each of the
line bundles
${V^{1}\coloneqq\mathcal{L}_{1}^{1}\oplus\ldots\oplus\mathcal{L}_{n}^{1}\rightarrow
X_{1}}$ and
$V^{2}\coloneqq\mathcal{L}_{1}^{2}\oplus\ldots\oplus\mathcal{L}_{n}^{2}\rightarrow
X_{2}$ are relatively orientable. Finally, the vector bundles $V_{1}$ and
$V_{2}$ are relatively orientable if and only the vector bundle $V$ is
relatively orientable since $\det V=p_{1}^{*}\det V^{1}\otimes p_{2}^{*}\det
V^{2}$ and $\omega_{X}=p_{1}^{*}\omega_{X_{1}}\otimes
p_{2}^{*}\omega_{X_{2}}$, so

$\det V\otimes\omega_{X}=p_{1}^{*}(\det V^{1}\otimes\omega_{X_{1}})\otimes
p_{2}^{*}(\det V^{2}\otimes\omega_{X_{2}})$

is a square if and only if $\det V^{1}\otimes\omega_{X_{1}}$ and $\det
V^{2}\otimes\omega_{X_{2}}$ are squares. ∎

######  Example 6.18.

Let $C_{1},C_{2},\dots,C_{n}$ be curves in $(\mathbb{P}^{1})^{n}$ defined by
$f_{i}$ of degree $(d_{1}^{i},d_{2}^{i},\dots,d_{n}^{i})$, for
$i=1,2,\dots,n$. Then $(f_{1},f_{2},\dots,f_{n})$ defines a section of

$V\coloneqq\bigoplus_{i=1}^{n}\mathcal{O}(d_{1}^{i},d_{2}^{i},\dots,d_{n}^{i})\rightarrow(\mathbb{P}^{1})^{n}$

where
$\mathcal{O}(d_{1}^{i},d_{2}^{i},\dots,d_{n}^{i})=\bigotimes_{j=1}^{n}\pi_{j}^{*}\mathcal{O}_{\mathbb{P}^{1}}(d_{j}^{i})$
and $\pi_{j}:(\mathbb{P}^{1})^{n}\rightarrow\mathbb{P}^{1}$ is the $j$th
projection. The vector bundle $V$ is relatively orientable if and only if

$\det
V\otimes\omega_{(\mathbb{P}^{1})^{n}}=\mathcal{O}\left(\sum_{i=1}^{n}d_{1}^{i}-2,\sum_{i=1}^{n}d_{2}^{i}-2,\dots,\sum_{i=1}^{n}d_{n}^{i}-2\right)$

is a square, which is the case if and only if $\sum_{i=1}^{n}d_{j}^{i}-2$ even
for every $j=1,2,\dots,n$. For every $i=1,2,\dots,n$, the Newton polygon
$\operatorname{NP}(f_{i})$ is the parallelepiped with a corner in $\bar{0}$
and side edges $d_{j}^{i}\mathrm{e}_{j}$, where
$\\{\mathrm{e}_{j}\\}_{j=1}^{n}$ is the standard basis. The Minkowski sum
$\sum_{i=1}^{n}\operatorname{NP}(f_{i})$ is the parallelepiped with a corner
in $\bar{0}$ and side edges $\sum_{i=1}^{n}d_{j}^{i}\mathrm{e}_{j}$. This
parallelepiped $\sum_{i=1}^{n}\operatorname{NP}(f_{i})$ has no odd vertices on
the boundary if and only if every $\sum_{i=1}^{n}d_{j}^{i},j=1,2,\dots,n$ is
even, that is exactly when $V$ is relatively orientable. Let
$\widetilde{C}_{i},i=1,2,\dots,n$ be enriched tropical curves with Newton
polytope equal to $\operatorname{NP}(f_{i})$ and assume that
$\sum_{i=1}^{n}d_{j}^{i},j=1,2,\dots,n$ is even. Then, Proposition 6.4 implies
that

$\displaystyle\sum_{p\in\bigcap_{i=1}^{n}C_{i}}\widetilde{\operatorname{mult}}_{p}(\widetilde{C}_{1},\widetilde{C}_{2},\dots,\widetilde{C}_{n})=\frac{1}{2}\left(\sum_{\sigma\in\mathcal{S}_{n}}d_{\sigma(1)}^{1}d_{\sigma(2)}^{2}\cdots
d_{\sigma(n)}^{n}\right)h=n^{\mathbb{A}^{1}}(V)\in\operatorname{GW}(k).$

Equivalently, this coincides with $n^{\mathbb{A}^{1}}(V)$ and yields an
enriched count of intersection points of $n$ curves in $(\mathbb{P}^{1})^{n}$.

For $(A_{1},\ldots,A_{n})$ combinatorially oriented, we get that the enriched
count of zeros of the system of equations $f_{1}=\ldots=f_{n}=0$ is
independent of the coefficients of $f_{1},\ldots,f_{n}$.

######  Theorem 6.19 (Enriched Bernstein-Kushnirenko theorem).

For a combinatorially oriented $n$-tuple of indexing sets
$(A_{1},\ldots,A_{n})$ and for a generic system of equations

$f_{1}(x)=\ldots=f_{n}(x)=0$

where $f_{i}\in L_{A_{i}}$, the enriched count of solutions in
$\operatorname{Spec}k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ equals

$\sum_{p}\operatorname{Tr}_{k(p)/k}\left\langle\det\operatorname{Jac}(f_{1},\ldots,f_{n})(p)\right\rangle)=\frac{\operatorname{MVol}(\Delta_{A_{1}},\ldots,\Delta_{A_{n}})}{2}h\in\operatorname{GW}(k).$

We can also say something about the non-orientable case. Just like in the case
of Bézout, then the enriched count depends on the choice of coefficients of
$f_{1},\ldots,f_{n}$. However, we still get a lower bound for the number of
hyperbolic summands depending on the number of odd points on the boundary of
$\Delta_{A_{1}}+\ldots+\Delta_{A_{n}}$.

######  Theorem 6.20.

Let $(A_{1},\ldots,A_{n})$ be a sequence of finite subsets of
$\mathbb{Z}^{n}$. Let

$N\coloneqq\operatorname{Card}(\partial(\Delta_{A_{1}}+\ldots+\Delta_{A_{n}})\cap\Lambda^{\text{odd}})$

be the number of odd points on the boundary of
$\Delta_{A_{1}}+\ldots+\Delta_{A_{n}}$. Then for a generic system of equations

$f_{1}(x)=f_{2}(x)=\ldots=f_{n}(x)=0$

where $f_{i}\in L_{A_{i}}$, we get that the enriched count of solutions in
$\operatorname{Spec}k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ equals

$\sum_{p}\operatorname{Tr}_{k(p)/k}\left\langle\det\operatorname{Jac}(f_{1},\ldots,f_{n})(p)\right\rangle=\frac{\operatorname{MVol}(\Delta_{A_{1}},\ldots,\Delta_{A_{n}})-r}{2}h+\left\langle
a_{1},\ldots,a_{r}\right\rangle\in\operatorname{GW}(k)$ (36)

for some $r\leq N$ and $a_{1},\ldots,a_{r}\in k^{\times}$.

###### Proof of Theorem 6.19 and Theorem 6.20.

Let

$f_{i}\coloneqq\sum_{I\in A_{i}}c_{i,I}x^{I}.$

We quadratically enrich the proof of [HS95]. Huber and Sturmfels look at toric
deformations

$f_{i,t}\coloneqq\sum_{I\in A_{i}}c_{i,I}x^{I}t^{\omega_{i}(I)}$

of the $f_{i}$ for a general choice $\omega_{i}(I)\in\mathbb{Q}$ and count the
number of algebraic functions $x(t)$ that are solutions to

$f_{1,t}=\ldots=f_{n,t}=0.$ (37)

Tropicalizing the $f_{i,t}$ yields enriched tropical hypersurfaces
$\widetilde{V}_{i}$. Huber and Sturmfels show that the number of algebraic
functions $x(t)$ that are solutions to the system (37) and tropicalize to a
point ${p\in\widetilde{V}_{1}\cap\ldots\cap\widetilde{V}_{n}}$ equals the
intersection multiplicity $\operatorname{mult}_{p}(V_{1},\ldots,V_{n})$.
Similarly, the enriched count of algebraic functions $x(t)$ that solve
$\eqref{eq:toric deformation}$ equals the sum of enriched intersection
multiplicities
$\widetilde{\operatorname{mult}}_{p}(\widetilde{V}_{1},\ldots,\widetilde{V}_{n})$
at the intersections points. Our Main Theorem 5.6 and Proposition 6.1 yields
the result. ∎

We provide examples where (36) is invariant, i.e., does not depend of the
coefficients of the $f_{i}$ and where it depends on the coefficients.

(a) $\operatorname{NP}(g)+\operatorname{NP}(h)$

(b) $\operatorname{NP}(g)+\operatorname{NP}(h^{\prime})$

Figure 12: Newton polygons

######  Example 6.21.

The following is the leading example in [Stu98] Let

$g(x,y)=a_{1}+a_{2}x+a_{3}xy+a_{4}y$

and

$h(x,y)=b_{1}+b_{2}x^{2}y+b_{3}xy^{2}.$

By Theorem 6.13 the number of solutions in
$\operatorname{Spec}\mathbb{C}[x^{\pm 1},y^{\pm 1}]$ to $g(x,y)=h(x,y)=0$ is
equals the mixed volume
$\operatorname{MVol}(\operatorname{NP}(g),\operatorname{NP}(h))$ of the Newton
polygons $\operatorname{NP}(g)$ and $\operatorname{NP}(h)$ of $g$ and $h$ for
a generic choice of coefficients $a_{1},a_{2},a_{3},a_{4},b_{1},b_{2},b_{3}$.
The enriched count of solutions to $g(x,y)=0$ and $h(x,y)=0$ depends on the
choice of coefficients since there are $2$ odd points on the boundary of
$\operatorname{NP}(g)+\operatorname{NP}(h)$ as shown the first picture in
Figure 12. For example if $k=\mathbb{R}$ and
$a_{1}=a_{2}=a_{3}=a_{3}=a_{4}=b_{1}=b_{2}=b_{3}=1$ we get that the enriched
count of zeros equals $2h$. However, if we set $b_{3}=-1$ (all other
coefficients are still equal to $1$), then the enriched count of zeros equals
$h+\left\langle 1,1\right\rangle$.

If we replace $h$ by $h^{\prime}(x,y)=b_{1}+b_{2}x^{3}+b_{3}x^{3}y$, the
Minkowski sum $\operatorname{NP}(g)+\operatorname{NP}(h^{\prime})$ of the
Newton polygons $\operatorname{NP}(g)$ and $\operatorname{NP}(h^{\prime})$ of
$g$ and $h^{\prime}$ has no odd points on the boundary as one can see in the
second picture of Figure 12. The mixed volume of $\operatorname{NP}(g)$ and
$\operatorname{NP}(h^{\prime})$ equals $4$, hence the enriched count of
solutions to $g(x,y)=h^{\prime}(x,y)=0$ in $\operatorname{Spec}(k[x^{\pm
1},y^{\pm 1}])$ equals $2h$ and this is independent of the choice of
coefficients.

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