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).