2 Quantization

2.1 Maslov dequantization of positive real numbers

The rescaling formulas $u_h=h\ln x$, $v_h=h\ln y$ bring to mind formulas related to the Maslov dequantization of real numbers, see e.g. [4], [5]. The core of the Maslov dequantization is a family of semirings $\left\{S_h\right\}_{h\in[0,\infty)}$ (recall that a semiring is a sort of ring, but without subtraction). As a set, each of $S_h$ is $\mathbb{R}$. The semiring operations $\oplus_h$ and $\odot_h$ in $S_h$ are defined as follows: for $h>0$ by

$$a\oplus_h b = h\ln(e^{a/h}+e^{b/h}),$$ (1)

$$a\odot_h b = a+b$$ (2)

and, for $h=0$, by

$$a\oplus_h b = \max\{a,b\},$$ (3)

$$a\odot_h b = a+b$$ (4)

These operations depend continuously on $h$. For each $h>0$ the map

$$D_h:\mathbb{R}_+\smallsetminus \{0\} \to S_h: x\mapsto h\ln x$$

is a semiring isomorphism of $\left\{\mathbb{R}_+\smallsetminus \{0\},+,\cdot\right\}$ onto $\left\{S_h,\oplus_h,\odot_h\right\}$, that is

$$D_h(a+b)=D_h(a)\oplus_hD_h(b),\qquad D_h(ab)=D_h(a)\odot_hD_h(b).$$

Thus $S_h$ with $h>0$ can be considered as a copy of $\mathbb{R}_+\smallsetminus \{0\}$ with the usual operations of addition and multiplication. On the other hand, $S_0$ is a copy of $\mathbb{R}$ where the operation of taking maximum is considered as an addition, and the usual addition, as a multiplication.

Applying the terminology of quantization to this deformation, we must call $S_0$ a classical object, and $S_h$ with $h\ne0$, quantum ones. The analogy with Quantum Mechanics motivated the following:

Correspondence Principle    formulated by Litvinov and Maslov [4]. ``There exists a (heuristic) correspondence, in the spirit of the correspondence principle in Quantum Mechanics, between important, useful and interesting constructions and results over the field of real (or complex) numbers (or the semiring of all nonnegative numbers) and similar constructions and results over idempotent semirings.''

This principle proved to be very fruitful in a number of situations, see [4], [5]. According to the correspondence principle, the idempotent counterpart of a polynomial $p(x)=a_nx^{n}+a_{n-1}x^{n-1}+\dots+a_0$ is a convex PL-function $M_p(u)=\max\left\{nu+b_n, (n-1)u+b_{n-1},\ \dots , b_0\right\}$. As we have seen above, $p$ and $M_p$ are related not only on an heuristic level. In Section 1.6 we connected the graph $\Gamma _p$ of $p$ on logarithmic paper and the graph $\Gamma _{M(p)}$ of $M_p$ by a continuous family of graphs $\{\Gamma ^h_{p_h}\}_{h\in(0,1)}$.

2.2 Logarithmic paper as a graphical device for the Maslov dequantization

As we saw in Section 1.5, the graph of a polynomial $p(x)=\sum_ka_kx^k$ with positive real coefficients $a_k=e^{b_k}$ on log paper is the graph of function $\mathbb{R}\to\mathbb{R}$ defined by $v=\ln\left(\sum_ke^{ku+b_k}\right)$. Observe that $\ln\left(\sum_ke^{ku+b_k}\right)$ is the value in $S_1$ of the polynomial $\sum_k b_kx^k$ at $x=u$. Therefore we can identify the graph $\Gamma _p$ of $p(x)=\sum_ka_kx^k$ on log paper with the (Cartesian) graph of the polynomial $\sum_k b_kx^k$ on $S_1^2$.

Furthermore, $\Gamma ^h_{p_h}$ is the graph of the function $\mathbb{R}\to\mathbb{R}$ defined by

$$v=h\ln\left(\sum_ka_k^{1/h}e^{(ku)/h}\right) =h\ln\left(\sum_ke^{(ku+b_k)/h} \right).$$

Observe, that the right hand side is the value in $S_h$ of the same polynomial $\sum_k b_kx^k$ at $u$. Therefore we can identify the graph $\Gamma ^h_{p_h}$ of $p_h(x)=\sum_ka_k^{1/h}x^k$ on log paper with the (Cartesian) graph of $\sum_k b_kx^k$ on $S_h^2$.

At last, the graph of $\sum_k b_kx^k$ on $S_0^2$ is the the graph of $M_p$.

We see that the whole job of deforming $\Gamma _p$ to the graph of a piecewise linear convex function can be done by the Maslov dequantization: the deformation consists of the graphs of the same polynomial $\sum_k b_kx^k$ on $S_h^2$ for $h\in[0,1]$. The coefficients $b_k$ of this polynomial are logarithms of the coefficients of the original polynomial: $b_k=\ln a_k$. Since the map $x\mapsto\ln x:\mathbb{R}_+\smallsetminus 0\to S_1$ was denoted above by $D_1$, we denote by $D_1 F$ the polynomial obtained from a polynomial $F$ with positive coefficients by replacing its coefficients with their logarithms. Thus $\sum_kb_kx^k=D_1p(x)$. Since $D_1$ is a semiring homomorphism, the graph $\Gamma _p$ of $p$ on log paper is the graph of $D_1p$ on $S_1^2$. The other graphs involved into the deformation are the graphs of the same polynomial $D_1p$ on $S_h^2$. They coincide with the graphs on log paper of the preimages $p_h$ of $D_1p$ under $D_h$. Indeed, $p_h(x)=\sum_ka_k^{1/h}x^k$ and $D_h^{-1}(b_k)=D_h^{-1}D_1(a_k)=e^{D_1(a_k)/h}=e^{(\ln a_k)/h}=a_k^{1/h}$.

For a real polynomial $p(x)=\sum_ka_kx^k$ with positive coefficients, we shall call $p_h(x)=\sum_ka^{1/h}_kx^k$ with $h>0$ the dequantizing family of polynomials.

2.3 Real algebraic geometry as quantized PL-geometry

The notion of polynomial is central in algebraic geometry. (I believe the subject of algebraic geometry would be better described by the name of polynomial geometry.) Since a polynomial over $\mathbb{R}$ is presented so explicitly as a quantization of a piecewise linear convex function, one may expect to find along this line explicit relations between other objects and phenomena of algebraic geometry over $\mathbb{R}$ and piecewise linear geometry. Indeed, in piecewise linear geometry the notion of piecewise linear convex function plays almost the same rôle as the notion of polynomial in algebraic geometry.

$$\begin{CD} \begin{matrix}\text{Real} \text{polynomials} \e... ...nd{matrix}@«\text{quantization}< \text{ PL-Geometry} \end{CD}$$

A representation of real algebraic geometry as a quantized PL-geometry may be rewarding in many ways. For example, in any quantization there are classical objects, i.e., objects which do not change much under the quantization. Objects of PL-geometry are easier to construct. If we knew conditions under which a PL object gives rise to a real algebraic object, which is classical with respect to the Maslov quantization, then we would have a simple way to construct real algebraic objects with controlled properties.