Dequantization of Real Algebraic Geometry on Logarithmic Paper

\[\begin {cases} u=\ln x\\ v=\ln y. \end {cases}\]

On a log paper the first quadrant is expanded homeomorphically to the whole plane, the line \(x=1\) occupies the position of the axis of ordinates, the line \(y=1\) occupies the position of the axis of abscissas, the unit square bounded by these lines and the coordinate axes occupies the whole third quadrant.

\[v=\ln y=\ln (1+x)=\ln (1+e^u).\]

See the left plot in Figure 1 , where the graph of \(v=\ln (1+e^u)\) is shown together with lines \(v=0\) and \(v=u\), which represent on the log paper the monomials \(1\) and \(x\) involved in our polynomial \(1+x\). The graph of \(v=\ln (1+e^u)\) looks like the broken line \(v=\max \{0,u\}\) with a smoothed corner: it goes along and above of this broken line getting very close to it as \(|u|\) grows. For \(|u|>4\) the difference becomes beyond the resolution.

Figure 1 Log paper graphs of \(1+x\),
\(1+e^{5}x+x^2\) and \(1+e^{-5}x+x^2\) .

Here are two further examples: the quadratic polynomials \(1+e^{\pm 5}x+x^2\). Then

\[v=\ln (1+e^{\pm 5}x+x^2)=\ln (1+e^{u\pm 5}+e^{2u}).\]

See the central and right plots in Figure 1 . The graph of \(v=\ln (1+e^{u\pm 5}+e^{2u})\) looks like the broken line \(v=\max \{0,u\pm 5,2u\}\) with smoothed corners. It goes along and above of this broken line getting very close to it far from its corners. Notice that the lines \(v=0\), \(v=u\pm 5\) and \(v=2u\) represent on the logarithmic paper the monomials \(1\), \(e^{\pm 5}x\) and \(x^2\), respectively.

\[L_p(u)=\ln \left (e^{nu+b_n}+e^{(n-1)u+b_{n-1}}+\dots +e^{b_0}\right )\]

and \(\GG _{M(p)}\) is the graph of a piecewise linear convex function

\[ M_p(u)=\max \left \{nu+b_n, \ (n-1)u+b_{n-1},\ \dots \ ,\ b_0 \right \}. \]

Obviously, \(M_p(u)\le L_p(u)\le M_p(u)+\ln (n+1)\). Hence \(\GG _{p}\) is above \(\GG _{M(p)}\), but below a copy of \(\GG _{M(p)}\) shifted upwards by \(\ln (n+1)\). The latter is in fact a rough estimate. It turns to equality only at \(u\), where all linear functions, whose maximum is \(M_p(u)\), are equal: \(nu+b_n= (n-1)u+b_{n-1}=\dots =b_0\).

For a generic value of \(u\), only one of these functions is equal to \(M_p(u)\). Say \(M_p(u)=ku+b_k\), while \(M_p(u)>d+lu+b_l\) for some positive \(d\) and each \(l\ne k\). Then

\[L_p(u)<M_p(u)+\ln (1+ne^{-d})<M_p(u)+e^{-d}n.\]

If for some value of \(u\) the values of all of the functions \(ku+b_k\) except two are smaller than \(M_p(u)-d\), then

\[L_p(u)<M_p(u)+\ln (2+(n-1)e^{-d})<M_p(u)+\ln 2 +e^{-d}(n-1)/2.\]

Thus, on a logarithmic paper the graph of a generic polynomial with positive coefficients lies in a narrow strip along the brocken line which is the graph of the maximum of its monomials. The width of the strip is estimated by characteristics of the mutual position of the lines which are the graphs of the monomials. The less congested the configuration of these lines, the norrower this strip.

A natural way to make a configuration of lines less congested without changing its topology is to apply a dilation \((u,v)\mapsto (Cu,Cv)\) with a large \(C>0\). In what follows it is more convenient to use instead of \(C\) a parameter \(h\) related to \(C\) by \(h=1/C\). In terms of \(h\) the dilation acts by \((u,v)\mapsto (u/h,v/h)\). It maps the graph of \(v=ku+b\) to the graph of \(v=ku+b/h\). The parallel operation on monomials replaces \(ax^{k}\) by \(a^{1/h}x^{k}\).

Consider the corresponding family of polynomials: \(p_{h}(x)=\sum _{k}a_k^{1/h}x^k\). On log paper, the graphs of its monomials are obtained by dilation with ratio \(1/h\) from the graphs of the corresponding monomials of \(p\). Hence \(\GG _{M(p_{h})}\) is the image of \(\GG _{M(p)}\) under the same dilation. However, \(\GG _{p_{h}}\) is not the image of \(\GG _p\). It still lies in a strip along \(\GG _{M(p_{h})}\) and the strip is getting narrower as \(h\) decreases, but at the corners of \(\GG _{M(p_{h})}\) the width of the strip cannot become smaller than \(\ln 2\).

To keep the picture of our expanding configuration of lines (the graphs of monomials) independent on \(h\), let us make an additional calibration of coordinates: set \(u_h=hu=h\ln x\), \(v_h=hv=h\ln y\). Denote by \(\GG ^h_f\) the graph of a function \(y=f(x)\) in the plane with coordinates \(u_h, v_h\).

Then \(\GG ^h_{M(p_{h})}\) does not depend on \(h\). The additional scaling reduces the width of the strip along \(\GG ^h_{M(p_h)}\), where \(\GG ^h_{p_h}\) lies, forcing the width to tend to 0 as \(h\to 0\). Thus \(\GG ^h_{p_{h}}\) tends to \(\GG ^h_{M(p_h)}\) (in the \(C^0\) sense) as \(h\to 0\).

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 \(\R \). The semiring operations \(\oplus _h\) and \(\odot _h\) in \(S_h\) are defined as follows:

\[D_h:\R _+\sminus \{0\} \to S_h: x\mapsto h\ln x\]

is a semiring isomorphism of \(\left \{\R _+\sminus \{0\},+,\cdot \right \}\) 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 \(\R _+\sminus \{0\}\) with the usual operations of addition and multiplication. On the other hand, \(S_0\) is a copy of \(\R \) 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 \(h\ne 0\), quantum ones. The analogy with Quantum Mechanics motivated the following correspondence principle formulated by Litvinov and Maslov [4] as follows:

“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 \(\GG _p\) of \(p\) on logarithmic paper and the graph \(\GG _{M(p)}\) of \(M_p\) by a continuous family of graphs \(\{\GG ^h_{p_h}\}_{h\in (0,1)}\).

Furthermore, \(\GG ^h_{p_h}\) is the graph of the function \(\R \to \R \) 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 \(\GG ^h_{p_h}\) of \(p_h(x)=\sum _ka_k^{1/h}x^k\) on log paper with the (Cartesian) graph of \(\sum _kb_kx^k\) 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 \(\GG _p\) 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 \(\sum _kb_kx^k\) 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 \(x\mapsto \ln x:\R _+\sminus 0\to S_1\) 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 \(\GG _p\) 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.

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 \(\R \) 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 \(\R \) 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.

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.

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 \(p(-x)\).

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 \(\GG _{p^+}\) and \(\GG _{p^-}\) 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 \(\GG _{p^+}\cap \GG _{p^-}\) 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 \(\GG _{M(p^+)}\) and \(\GG _{M(p^-)}\) 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.

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 \(p^{\Ge }\) with \(\Ge =\pm \) the sum of monomials corresponding to the planes of faces forming \(U^{\Ge }\) and put \(p=p^+-p^-\).

Consider now for \(1\ge h\ge 0\) 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 \(\GG _{M(p^+)}\), \(\GG _{M(p^-)}\). 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 \(\p U^+\) to the \((u,v)\)-plane.

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.

• we restrict to the case of nonsingular planar curves,
• we assume that all patches are trinomials, and
• we consider only the part of the curve contained in the first quadrant (what happens in other quadrants is described soon after).

Initial data. Let \(m\) be a positive integer (it will be the degree of the curve under construction) and \(\GD \) be the triangle in \(\R ^2\) with vertices \((0,0)\), \((m,0)\), \((0,m)\). Let \(\tau \) be a convex triangulation of \(\GD \) with vertices having integer coordinates. The convexity of \(\tau \) means that there exists a convex piecewise linear function \(\nu :\GD \longrightarrow {\R _+}\) which is linear on each triangle of \(\tau \) and is not linear on the union of any two triangles of \(\tau \). Let the vertices of \(\tau \) be equipped with signs. The sign (plus or minus) at the vertex with coordinates \((k,l)\) is denoted by \(\Gs _{k,l}\).

Construction of the piecewise linear curve. If a triangle of the triangulation \(\tau \) has vertices of different signs, draw a midline separating pluses from minuses. Denote by \(L\) the union of these midlines. It is a collection of polygonal lines contained in \(\GD \). The pair \((\GD , L)\) is called the result of combinatorial patchworking.

Construction of polynomials. Given initial data \(m\), \(\GD \), \(\tau \) and \(\Gs _{k,l}\) as above and a positive convex function \(\nu \) certifying, as above, that the triangulation \(\tau \) is convex. Consider a one-parameter family of polynomials

Patchwork Theorem. Let \(m\), \(\GD \), \(\tau \), \(\Gs _{k,l}\) and \(\nu \) be initial data as above. Denote by \(b_t\) the polynomials obtained by the polynomial patchworking of these initial data, and by \(L\) the PL-curve in \(\GD \) obtained from the same initial data by combinatorial patchworking.

Then for all sufficiently small \(t>0\) the polynomial \(b_t\) defines in the first quadrant \(\R ^2_{++}=\{(x,y)\in \R ^2\mid x,y>0\}\) a curve \(a_t\) such that the pair \((\R ^2_{++},\; a_t)\) is homeomorphic to the pair \((\Int \GD ,\;L\cap \Int \GD )\).

The Patchwork Theorem applied to \(b_t(-x,y)\), \(b_t(x,-y)\) and \(b_t(-x,-y)\) gives a similar topological description of the curve defined in the other quadrants by \(b_t\) with sufficiently small \(t>0\). The results can be collected in the following natural combinatorial construction.

Construction of the PL-curve. Take copies \(\GD _{x} = s_x(\GD )\), \(\GD _{y} = s_y(\GD )\), \(\GD _{xy} = s(\GD )\) of \(\GD \), where \(s_x,\; s_y\) are reflections with respect to the coordinate axes and \(s= s_x \circ s_y\). Denote by \(A\GD \) the square \(\GD \cup \GD _x \cup \GD _y \cup \GD _{xy}\). Extend the triangulation \(\tau \) to a symmetric triangulation of \(A\GD \), and the distribution of signs \(\Gs _{i,j}\) to a distribution at the vertices of the extended triangulation by the following rule: \(\Gs _{i,j}\Gs _{\Ge i,\Gd j}\Ge ^i\Gd ^j=1\), where \(\Ge ,\Gd =\pm 1\). In other words, passing from a vertex to its mirror image with respect to an axis we preserve its sign if the distance from the vertex to the axis is even, and change the sign if the distance is odd.

If a triangle of the triangulation of \(A\GD \) has vertices of different signs, select (as above) a midline separating pluses from minuses. Denote by \(AL\) the union of the selected midlines. It is a collection of polygonal lines contained in \(A\GD \). The pair \((A\GD , AL)\) is called the result of affine combinatorial patchworking. Glue by \(s\) the sides of \(A\GD \). The resulting space \(P\GD \) is homeomorphic to the real projective plane \(\rpp \). Denote by \(PL\) the image of \(AL\) in \(P\GD \) and call the pair \((P\GD ,PL)\) the result of projective combinatorial patchworking.

Addendum to the Patchwork Theorem. Under the assumptions of Patchwork Theorem above, for all sufficiently small \(t>0\) there exist a homeomorphism \(A\GD \to \R ^2\) mapping \(AL\) onto the the affine curve defined by \(b_t\) and a homeomorphism \(P\GD \to \R P^2\) mapping \(PL\) onto the projective closure of this affine curve.

The polynomial \(b_t\) defined by ( 3 ) is presented as \(b_t^+-b_t^-\), where

\[b_t^{\Ge }(x,y)=\sum _{\text {\scriptsize $\begin {aligned}&{ \text {$(k,l)$ runs over the vertices}}\\ &\text {\quad of } \tau , \text { at which $\Gs _{k,l}=\Ge $ }\end {aligned}$}}t^{\nu (k,l)}x^ky^l\]

Observe that polynomials \(b_t^{\pm }\) comprise dequantizing families. Indeed, if we take \(p(x,y)=\sum _{k,l}a_{k,l}x^ky^l\) with \(a_{k,l}=e^{-\nu (k,l)}\), then for \(h=-1/{\ln t}\) we obtain

\begin{multline*} p_h(x,y)=\sum _{k,l}a_{k,l}^{1/h}x^ky^l=\sum _{k,l}e^{-\nu (k,l)/h}x^ky^l=\\ \sum _{k,l}e^{\nu (k,l)\ln t}x^ky^l=\sum _{k,l}t^{\nu (k,l)}x^ky^l. \end{multline*} Patchwork Theorem deals with sufficiently small positive \(t\), while \(h\) in a dequantizing family of polynomials was a small positive number approaching 0. This is consistent with our setting \(h=-1/\ln t\).

A monomial \(a_{k,l}x^ky^l=e^{-\nu (k,l)}x^ky^l\) is presented in the logarithmic space by the graph of \(w=ku+lv-\nu (k,l)\). Hence the graph of the maximum of linear forms corresponding to all monomials of \(p^+\) and \(p^-\) is defined by

Some of these faces correspond to monomials of \(b_t^+\), the others to monomials of \(b_t^-\). The edges which separate the faces of these two kinds constitute a broken line as in Section 3.4 . These edges are dual to the edges of \(\tau \) which intersect the result \(L\) of the combinatorial patchworking.

Therefore the topology of the projection to the \((u,v)\)-plane of the broken line coincides with the topology of \(L\) in \(\GD \).

We see that the quantum point of view (or its graphical log paper equivalent) gives a natural explanation to the simplest patchwork construction. The proofs become more conceptual and straight-forward. Of course, similar but slightly more involved quantum explanations can be given to all versions of patchwork.

Let me shortly mention other problems which can be attacked using similar arguments.

First of all, this is the Fewnomial Problem. Although A. G. Khovansky [3] proved that basically all topological characteristics of a real algebraic variety can be estimated in terms of the number of monomials in the equations, the known estimates seem to be far weaker than conjectures. For varieties classical from the quantum point of view a strong estimates are obvious. It is very compelling to estimate how much the topology can be complicated by the quantizing deformations.

There seem to be deep relations between the dequantization of algebraic geometry considered above and the results of I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky on discriminants [1] . In particular, some monomials in a discriminant are related to intersections of hyperplanes in the dequantized polynomial.

Complex algebraic geometry also deserves a dequantization. Especially relevant may be amoebas introduced in [1] .