3 Algebraic geometry on logarithmic paper

3.1 Speak of real algebraic geometry only positively

Above in Section 1 we discussed the drawing of graphs on a logarithmic paper only for a polynomial with positive coefficients. The graphs allowed us to see the behavior of the polynomials only at positive values of the argument. This was for a good reason: we used logarithms of coordinates. The Maslov dequantization deals only with positive numbers. Therefore each fragment of algebraic geometry that we want to dequantize must be reformulated first only in terms of positive numbers. This seems to be possible for everything belonging to real algebraic geometry.

3.2 Visualizing roots of a polynomial on logarithmic paper

Above we could not encounter roots of polynomials, for a polynomial with positive coefficients has no positive roots. However if we really want to do algebraic geometry on log paper, we must figure out how to use graphs on log paper for visualizing roots (well, only positive roots) of an arbitrary real polynomial.

Any real polynomial $p(x)$ is a difference $p^+(x)-p^-(x)$ of polynomials with positive coefficients. We can reformulate the problem of finding the positive roots of $p$ as the problem of finding positive values of $x$ at which $p^+(x)=p^-(x)$. The graphs of $p^+$ and $p^-$ can be drawn on a log paper, where they are localized in the strips along broken lines, see Section 1.5 above. For some polynomials this picture gives a decent information on the number and position of the positive roots.

The negative roots of $p(x)$ can be treated in the same way, since their absolute values are the positive roots of $p(-x)$.

3.3 Plane algebraic curves on logarithmic paper

Now consider a real polynomial $p(x,y)=\sum_{k,l}a_{k,l}x^ky^l$ in two variables. Similarly to the case of polynomials in one variable, in the logarithmic space the graph of a monomial $ax^ky^l$ with $a>0$ is a plane $w=ku+lv+\ln a$, and the graph $\Gamma _p$ of a polynomial $p(x,y)$ with positive coefficients lies in a neighborhood of a convex piecewise linear surface, which is the graph $\Gamma _{M(p)}$ of the maximum $M(p)$ of the monomials. Furthermore, $p$ is included into a dequantizing family $p_h$ defined as $\sum_{k,l}a_{k,l}^{1/h}x^ky^l$ for $h>0$, cf. Section 2.2. The graph $\Gamma ^h_{p_h}$ of $p_h$ in the logarithmic space with scaled coordinates $u_h=h\ln x$, $v_h=h\ln y$, $w_h=h\ln z$ coincides with the graph of the polynomial $D_1 p(x,y)=\sum_{k,l}\left(\ln a_{k,l}\right)x^ky^l$ in $S_h^3$. These graphs with $h\in(0,1]$ constitute a continuous deformation of $\Gamma _p=\Gamma ^1_{p_1}$ to $\Gamma _{M(p)}$.

For a polynomial $p$ in two variables with arbitrary real coefficients, denote by $p^+$ the sum of its monomials with positive coefficients, and put $p^-=p^+-p$. Thus $p$ is canonically presented as a difference $p^+-p^-$ of two polynomials with positive coefficients. To obtain the curve defined on logarithmic paper by the equation $p(x,y)=0$, one can construct the graphs $\Gamma _{p^+}$ and $\Gamma _{p^-}$ for $p^+$ and $p^-$ in the logarithmic space, which are the surfaces defined in the usual Cartesian coordinates by $w=\ln\left(p^{\pm}(e^u,e^v)\right)$, and project the intersection $\Gamma _{p^+}\cap\Gamma _{p^-}$ to the plane of arguments.

For the first approximation of this curve, one may take the broken line, which is the projection of the intersection of the piecewise linear surfaces $\Gamma _{M(p^+)}$ and $\Gamma _{M(p^-)}$ corresponding to $p^+$ and $p^-$.

Of course, it may well happen that the broken line does not even resemble the curve. This happens to first approximations. However, it is very appealing to figure out circumstances under which the broken line is a good approximation, for a broken line seems to be much easier to deal with than an algebraic curve.

3.4 Constructing algebraic curves, which are classical from our quantum point of view

Recall that in the logarithmic space the graph of $ax^ky^l$ is a plane $w=ku+lv+\ln a$. It has a normal vector $(k,l,-1)$ and intersects the vertical axis at $(0,0,\ln a)$. Thus if we want to construct a curve of a given degree $m$, we have to arrange planes whose normals are fixed: they are $(k,l,-1)$ with integers $k,l$, satisfying inequalities $0\le k,l,k+l\le m$. The only freedom is in moving them up and down.

Consider the pieces of these planes which do not lie under the others. They form a convex piecewise linear surface $U$, the graph of the maximum of the linear forms defining our planes. The combinatorial structure of faces in $U$ depends on the arrangement. Assume that at each vertex of $U$ exactly three of the planes meet. This is a genericity condition, which can be satisfied by small shifts of the planes.

Divide now the faces of $U$ arbitrarily into two classes. Denote the union of one of them by $U^+$, the union of the other by $U^-$. By genericity of the configuration, the common boundary of $U^+$ and $U^-$ is union of disjoint polygonal simple closed curves. It can be easily realized as the intersection of PL-surfaces $\Gamma _{M(p^+)}$ and $\Gamma _{M(p^-)}$ as above: take for $p^{\varepsilon }$ with $\varepsilon =\pm$ the sum of monomials corresponding to the planes of faces forming $U^{\varepsilon }$ and put $p=p^+-p^-$.

Consider now for $1\ge h\ge0$ the curve $C_h\subset S_h^3$ which is the intersection of the graphs in $S_h^3$ of the polynomials $D_1 p^+$ and $D_1p^-$. At $h=0$ this is the intersection of the convex PL-surfaces $\Gamma _{M(p^+)}$, $\Gamma _{M(p^-)}$. Due to the genericity condition above, this intersection is as transversal as one could wish: at all but a finite number of points the interior part of a face of one of them meets the interior part of a face of the other one, and at the rest of the points an edge of one of the surfaces intersects transversaly the interior of a face of the other surface.

When $h$ gets positive, the graphs are smoothed, their corners are rounded off. The same happens to their intersection curve. While the graphs are transveral, the intersection curve is deformed isotopically.

Take the curve corresponding to a value of $h$ such that the transversality is preserved between 0 and this value. The projection to $(u,v)$-plane of $C_h$ represents an algebraic curve of degree $m$ on the scaled logarithmic paper and it can be obtained by a small isotopy of the projection of $\partial U^+$ to the $(u,v)$-plane.

3.5 Is this a patchworking?

A construction, which looks similar, has been known in the topology of real algebraic varieties for about 20 years as patchworking, or Viro's method. It has been used to construct real algebraic varieties with controlled topology and helped to solve a number of problems. For example, to classify up to isotopy non-singular real plane projective curves of degree 7 [7], [8] and disprove the Ragsdale conjecture [2] on the topology of plane curves formulated [6] as early as in 1906. To the best of my knowledge, the patchworking has never been related to the Maslov quantization.