Let $v:\Delta\cap\mathbb{Z}^{n+1}\to\mathbb{R}$ be any function and $a(z)=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}a_{j}z^{j}$ be any polynomial. Following [15] we define the patchworking polynomial for any $t>0$ by

$$f^{v}_{t}(z)=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}a_{j}t^{-v(j)}z^{j},$$

where $a_{j}\neq 0$ for any $j\in\Delta\cap\mathbb{Z}^{n+1}$. Note that if $v$ is integer-valued then $f^{v}_{t}$ makes sense also for any $t\in\mathbb{C}^{*}$.

If $V\subset(\mathbb{C}^{*})^{n+1}$ be an algebraic variety. The image $\operatorname{Log}(V)\subset(\mathbb{C}^{*})^{n+1}$ is called the amoeba of $V$, see [4]. Note that amoebas make sense also for varieties over other fields $K$ as long as we have a norm $K^{*}=K\smallsetminus\{0\}\to\mathbb{R}_{+}$. The map $\operatorname{Log}_{K}:(K^{*})^{n+1}\to\mathbb{R}^{n+1}$ is defined by $\operatorname{Log}_{K}(z_{1},\dots,z_{n+1})=(\log||z_{1}||_{K},\dots,\log||z_{n+1}||)$ and the amoeba of $V_{K}\subset(K^{*})^{n+1}$ is defined to be $\operatorname{Log}_{K}(V_{K})$.

A particularly useful case is when $K$ is an algebraically closed field with a non-Archimedian valuation. Recall that a non-Archimedian valuation is a function $\operatorname{val}:K^{*}\to\mathbb{R}$ ²² 2 Sometimes a valuation is defined as minus such a function. such that $\operatorname{val}(a+b)\leq\max\{\operatorname{val}(a),\operatorname{val}(b)\}$ and $\operatorname{val}(ab)=\operatorname{val}(a)+\operatorname{val}(b)$. Note that $e^{\operatorname{val}}$ gives a norm on $K$ and $\operatorname{Log}_{K}$ is nothing but taking the coordinatewise valuation.

Non-Archimedian amoebas of hypersurfaces were completely described in [7]. An example of such field is the field $K$ of the Puiseux series with complex coefficients in $t$. Namely an element of $K$ is a formal series $b(t)=\sum\limits_{k\in J}b_{k}t^{k}$, $b_{k}\in\mathbb{C}^{*}$ where $J\subset\mathbb{R}$ is any bounded from below set contained in a finite union of arithmetic progressions. The valuation is defined by $\operatorname{val}||b(t)||=-\min J$. Note that we used irrational as well as rational powers in the Puiseux series to make the valuation surjective.

Consider the map $u:K^{*}\to S^{1}$ defined by $u(b)=\arg(b_{-\operatorname{val}(b)})$, $b=\sum\limits_{k\in J}b_{k}t^{k}$. In other words, $u$ takes the argument of the coefficient at the lowest power of $t$. This is a homomorphism from the multiplication group $K^{*}$. Together with $\operatorname{val}$ it gives a homomorphism $w=(\operatorname{val},u):K^{*}\to\mathbb{C}^{*}\approx\mathbb{R}\times S^{1}$ and thus a homomorphism $W:(K^{*})^{n+1}\to(\mathbb{C}^{*})^{n+1}$.

Consider the following family of binary operations on $\mathbb{R}\ni x,y$:

$$x\oplus_{t}y=\log_{t}(t^{x}+t^{y}),$$

for $t>1$ and

$$x\oplus_{\infty}y=\lim\limits_{t\to 0}x\oplus_{t}y=\max\{x,y\}.$$

This is a commutative semigroup operation (no inverse elements and no zero) for each $t$. The set $\mathbb{R}$ equipped with this operation for addition and with $x\odot y=x+y$ for multiplication is a semiring $\mathbb{R}_{t}$. Indeed, for any $x,y,z\in\mathbb{R}$ we have $x\odot(y\oplus_{t}z)=(x\odot z)\oplus_{t}(y\odot_{t}z)$.

Passing from a finite $t$ to infinity in this family of semirings is called Maslov’s dequantization, cf. [10]. Note that for all finite values of $t$ the semiring is isomorphic to the semiring of real positive numbers equipped with the usual addition and multiplication. But the behavior at $t=\infty$ is qualitatively different, the addition becomes idempotent, $x\oplus_{\infty}x=x$. The prefix “de” reflects the fact that in this deformation the classical calculus operations appear on the quantum side.

There is a universal bound for the convergence of the operations $\oplus_{t}$ to $\oplus_{\infty}=\max$. Namely, we have

(4)

$$\max\{x_{1},\dots,x_{N}\}\leq x_{1}\oplus_{t}\dots\oplus_{t}x_{N}\leq\max\{x_{1},\dots,x_{N}\}+\log_{t}N$$

The dequantization point of view can be used to reinterpret Viro’s patchworking, see [16]. Instead of deforming the coefficients of the polynomial we may keep them constant, but deform the addition operation instead. This point of view yields some useful estimates on the zero set of the patchworking polynomial as shown below.

One way to think of a polynomial is to think of it as a collection of coefficients at its monomials. Fix a polynomial $p(x)={\sum\limits_{j}}c_{j}x^{j}$ in $n+1$ variables, where the arithmetic operations are taken from the semiring $\mathbb{R}_{t}$. Depending on $t$ this polynomial defines different functions $p_{t}:\mathbb{R}^{n+1}\to\mathbb{R}$. Note that the function

$$f_{t}(z)=t^{p_{t}(\operatorname{Log}_{t}(z))}$$

coincides with the patchworking polynomials where all $a_{j}=1$ and $v(j)=c_{j}$. Here $\operatorname{Log}_{t}(z_{1},\dots,z_{n+1})=(\log_{t}(z_{1}),\dots,\log(z_{n+1})).$

Let $f_{t}=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}a_{j}t^{-v(j)}z^{j}$ now be a general patchworking polynomial. Denote ${V}^{\circ}_{t}=\{f_{t}=0\}\subset(\mathbb{C}^{*})^{n+1}$. The family $f_{t}$ can be treated as a single polynomial in $(K^{*})^{n+1}$ (see 5.2). It defines a hypersurface ${V}^{\circ}_{K}\subset(K^{*})^{n+1}$. Consider the Hausdorff metric on the subsets of $\mathbb{R}^{n+1}$ induced by the Euclidean metric on $\mathbb{R}^{n+1}$. Denote $\mathcal{A}_{t}=\operatorname{Log}_{t}({V}^{\circ}_{t})$ and $\mathcal{A}_{K}=\operatorname{Log}_{K}({V}^{\circ}_{K})$.

This corollary can be strengthened to describe the limits of the varieties ${V}^{\circ}_{t}\subset(\mathbb{C}^{*})^{n+1}$ under the corresponding renormalization of the norms of their points. The description is in terms of the lifts of non-Archimedian amoebas, see 5.3. Let $H_{t}:(\mathbb{C}^{*})^{n+1}\to(\mathbb{C}^{*})^{n+1}$ be the transformation defined by

$$H_{t}(z_{1},\dots,z_{n+1})=(t^{-|z_{1}|}\frac{z_{1}}{|z_{1}|},\dots,t^{-|z_{n+1}|}\frac{z_{n+1}}{|z_{n+1}|}).$$

We have $\operatorname{Log}_{t}=\operatorname{Log}\circ H_{t}$.

The proof is the same as the proof of Corollary 5.4. The only difference we have to make is to incorporate the arguments of the monomials to the inequalities (5).

Let $\Pi$ be a maximal dual $\Delta$-complex and $v:\Delta\cap\mathbb{Z}^{n+1}\to\mathbb{R}$ be the function such that $\Pi=\Pi_{v}$ as in Proposition 1.4. It gives us a patchworking polynomial $f_{t}=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}t^{-v(j)}z^{j}$. As before we denote with ${V}^{\circ}_{t}\subset(\mathbb{C}^{*})^{n+1}$ the zero set of this polynomial.

We construct $\lambda_{t}:{V}^{\circ}_{t}\to\Pi$ for a sufficiently large $t$ by gluing the fibrations $\lambda_{H}$ from 3.3.

To do it we construct a singular foliation $\mathcal{F}_{\Pi}$ in a neighborhood $\mathcal{N}\supset\Pi$. By Proposition 1.11 $\Pi$ can be locally identified with $\Sigma_{n}$ by elements of $ASL_{n+1}(\mathbb{Z})$. Recall that an element $M\in ASL_{n+1}(\mathbb{Z})$ is a rotation defined by a unimodular integer $(n+1)\times(n+1)$-matrix $(m_{j,k})$ followed by a translation by $m=(m_{1},\dots,m_{n+1})$ in $\mathbb{R}^{n+1}$. This transformation of $\mathbb{R}^{n+1}$ lifts to $(\mathbb{C}^{*})^{n+1}$ as

$$H_{M}:z_{j}\mapsto m_{j}z_{1}^{m_{j,1}}\dots z_{n+1}^{m_{j,n+1}}.$$

We patch the foliations $\mathcal{F}$ constructed in 3.3 for the primitive $n$-complex $\Sigma_{n}$. Let $v_{j}\in\Pi$ be a vertex. By Proposition 1.11 there exists a neighborhood $U_{j}\ni v_{j}$ in $\Pi$ and $M_{j}\in ASL_{n+1}(\mathbb{Z})$ such that $M_{j}(U_{j})$ is a neighborhood of $0$ in $\Sigma_{n}$. Let $N_{j}$ be a small neighborhood of the closure of $M_{j}(U_{j})$.

Consider the pull-back under $M_{j}$ of the foliation $\mathcal{F}$ constructed in 3.3 restricted to $N_{j}$. Note that $M_{j}^{-1}(N_{j})$ cover $\Pi$. The pull-back foliations at the overlaps $M_{j}^{-1}(N_{j})\cap M_{k}^{-1}(N_{k})$ do agree in general. Nevertheless, they have the same type of singularities at the same points and their non-singular leaves are transverse to $\Pi$. A partition of unity gives a foliation $\mathcal{F}_{\Pi}$ in a neighborhood $\mathcal{N}$ of $\Pi$. Note that we can ensure that $\mathcal{N}$ contains an $\epsilon$-neighborhood of $\Pi$ for some $\epsilon>0$. Following 3.3 we denote $\pi_{\mathcal{F}_{\Pi}}:\mathcal{N}\to\Pi$ the projection along the leaves of $\mathcal{F}_{\Pi}$.

By Corollary 5.4 for a sufficiently large $t>0$ we have $\operatorname{Log}_{t}(V_{t})\subset\mathcal{N},$ and we define

$$\lambda_{t}=\pi_{\mathcal{F}_{\Pi}}\circ\operatorname{Log}_{t}:V_{t}\to\Pi.$$

Here we prove that $V_{t}$ is non-singular and that $\lambda_{t}$ satisfies to all hypotheses of Theorem 3 for a large $t>0$.

Note that $||t^{-v(j)}z^{j}||=t^{jx-v(j)}$, where $jx\in\mathbb{R}$ stands for the scalar product, at any $z$ such that $\operatorname{Log}_{t}=x$. Let $F\subset\Pi$ be an open $(n+2-k)$-cell.

This lemma implies that ${V}^{\circ}_{t}$ is non-singular for large $t>0$. Indeed, it is covered by a finite number of open sets and in each set it is a small perturbation of a hyperplane. Furthermore, its compactification $V_{t}\subset\mathbb{C}T_{\Delta}$ is smooth and transverse to the coordinate hyperplanes as the same reasoning with the terms of smaller order applies to the affine charts of $\mathbb{C}T_{\Delta}$.

Our next step is to isotop $V_{t}$ over $N_{j}$ as in 3.4. Recall that $N_{j}$ was defined in 5.5 as a small neighborhood of $\bar{U_{j}}\subset\Pi$ in $\mathbb{R}^{n+1}$. Denote

$$Q^{n}_{j}=M_{j}^{-1}(H_{t}(Q^{n}))\cap\operatorname{Log}_{t}^{-1}(N_{j}).$$

By the last conclusion of Proposition 3.6 these manifolds coincide over $N_{j}\cap N_{k}$ for $t>>0$. We set

$$Q_{\Pi}=\bigcup\limits_{j}Q^{n}_{j}.$$

Note that for $t>>0$ ${V}^{\circ}_{t}$ is isotopic to $Q_{\Pi}$ by the same isotopy as in the proof of Proposition 3.6 since all other monomials of $f_{t}$ have smaller order in $t$. This proves Theorem 4. As in Proposition 3.6 the closure $\bar{Q_{\Pi}}\subset\mathbb{C}T_{\Delta}$ is a smooth manifold. Similarly, $V_{t}$ is isotopic to $\bar{Q_{\Pi}}$ in $\mathbb{C}T_{\Delta}$.

In the proof of Theorem 3 we may assume that ${V}^{\circ}={V}^{\circ}_{t}$ since its closure $V_{t}\subset\mathbb{C}T_{\Delta}$ is smooth and transverse to the coordinate hyperplanes. Similarly, in the proof of Theorems 1, 1’ and 2 we may assume that $V=V_{t}$. We define $\lambda^{\circ}:{V}^{\circ}\to\Pi$ as a composition of the isotopy $V\approx Q_{\Pi}$ (note that since this map is realized by an ambient isotopy it is a symplectomorphism by Moser’s trick), the map $\operatorname{Log}_{t}:(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}$ and the projection $\pi_{\mathcal{F}_{\Pi}}:\mathcal{N}\to\Pi$. To define $\lambda:V\to\bar{\Pi}$ we compactify the previous construction by using $\bar{Q_{\Pi}}$ and the reparametrized moment map to $\Delta$ as in 1.3.

This proves Theorem 2, since everything in our construction is equivariant with respect to complex conjugation as long as $a_{j}$ in the patchworking polynomial are real. The fibration $\lambda$ is totally real since it is totally real for a hyperplane.

Also, by Proposition 3.6 this proves the second and the third conclusions in Theorems 1, 1’ and 3. The homotopy type of $\bar{\Pi}$ and $\Pi$ is the wedge of $p_{g}$ copies of $S^{n}$, where $p_{g}=h^{n,0}$ by Proposition 1.10.

To finish the proof of Theorems 1. 1’ and 3 we need to prove injectivity of the induced homomorphism in cohomology and to exhibit the Lagrangian spheres lifting the cycles from $\Pi$.

The Lagrangian spheres will come from components of certain real hypersurfaces whose complexification is isotopic to $V$.

Let $j$ be a lattice point of $\Delta$. We define

$$f_{t}^{(j)}=\sum\limits_{k\neq j}|a_{k}|t^{v(k)}z^{k}-|a_{j}|t^{v(j)}z^{j}.$$

Denote with $V_{t}^{(j)}\subset(\mathbb{C}^{*})^{n+1}$ the zero set of $f_{t}^{(j)}$ and with $\mathbb{R}V_{t}^{(j)}\subset(\mathbb{R}^{*})^{n+1}$ its real part. The Viro patchworking theorem [15] (see also [4] for a special case of combinatorial patchworking and [5] for an elementary description in the case of curves) implies that $\mathbb{R}V_{t}^{(j)}\cap\mathbb{R}_{+}^{n+1}$ is diffeomorphic to a sphere $S^{n}$. This sphere $S^{n}_{j}\subset V_{t}^{(j)}$ is Lagrangian as a component of the real part and it maps under $\operatorname{Log}_{t}$ to $\mathcal{N}\supset\Pi$ for $t>>0$. Furthermore, it realizes in $H_{n}(\Pi)$ the class corresponding to $j$ according to Proposition 1.10.

By 5.6 $V_{t}^{(j)}$ is smooth. Thus, it is isotopic to $V_{t}$ and we have a diffeomorphism $h:V_{t}^{(j)}\to V_{t}$. Moreover, we can choose an isotopy among the hypersurfaces defined by such polynomials that the norm of all monomials is constant in the course of deformation. All such hypersurfaces are smooth and their image under $\operatorname{Log}_{t}$ is contained in $\mathcal{N}\supset\Pi$ by 5.6. Therefore, the image $h(S^{n}_{j})$ projects to the same class in $H_{n}(\Pi)$.

By Moser’s trick, $h$ is isotopic to a symplectomorphism. This gives a Lagrangian sphere in $V_{t}$ which projects to the class in $H_{n}(\Pi)$ corresponding to $j$. Thus the last conclusion of Theorems 1 and 1’ is proved.

Existence of such spheres also implies the first conclusion of Theorems 1 and 1’. The map $\lambda^{*}$ is injective since we can distinguish the images in $H^{n}(V;\mathbb{Z})$ by their evaluations on these Lagrangian spheres.

The proof of Theorem 3 is the same since these spheres belong to the toric part $\mathbb{R}{V}^{\circ}_{t}$ of $\mathbb{R}V$.