4 Patchworking real algebraic curves

4.1 The simplest patchworking

Here is a description of a simplified version of patchworking. The simplifications are of the following 3 kinds:

• 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 $\Delta$ be the triangle in $\mathbb{R}^2$ with vertices $(0,0)$, $(m,0)$, $(0,m)$. Let $\tau$ be a convex triangulation of $\Delta$ with vertices having integer coordinates. The convexity of $\tau$ means that there exists a convex piecewise linear function $\nu:\Delta \longrightarrow {\mathbb{R}_+}$ 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 $\sigma _{k,l}$.

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 $\Delta$. The pair $(\Delta , L)$ is called the result of combinatorial patchworking.

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

$$b_t(x,y)=\sum_{\text{\scriptsize$\begin{aligned}&{ \text{$(k,l)... ...rtices of $\tau$}\end{aligned}$}}\sigma _{k,l}t^{\nu(k,l)}x^ky^l.$$

The polynomials $b_t$ are called the results of polynomial patchworking.

Patchwork Theorem. Let $m$, $\Delta$, $\tau$, $\sigma _{k,l}$ 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 $\Delta$ 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 $\mathbb{R}^2_{++}=\{(x,y)\in\mathbb{R}^2\mid x,y>0\}$ a curve $a_t$ such that the pair $(\mathbb{R}^2_{++},\; a_t)$ is homeomorphic to the pair $(\operatorname{Int}\Delta ,\;L\cap\operatorname{Int}\Delta )$.

Figure 3: Patchworking: initial data, construction of the PL-curve in the first quadrant and on the whole plane. The corresponding algebraic curves are ellipses meeting the coordinate axes in their positive halves.

4.2 Patchwork in other quadrants

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 $\Delta _{x} = s_x(\Delta )$, $\Delta _{y} = s_y(\Delta )$, $\Delta _{xy} = s(\Delta )$ of $\Delta$, where $s_x,\; s_y$ are reflections with respect to the coordinate axes and $s= s_x \circ s_y$. Denote by $A\Delta$ the square $\Delta \cup \Delta _x \cup \Delta _y \cup \Delta _{xy}$. Extend the triangulation $\tau$ to a symmetric triangulation of $A\Delta$, and the distribution of signs $\sigma _{i,j}$ to a distribution at the vertices of the extended triangulation by the following rule: $\sigma _{i,j}\sigma _{\varepsilon i,\delta j}\varepsilon ^i\delta ^j=1$, where $\varepsilon ,\delta =\pm1$. 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 $A\Delta$ 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 $A\Delta$. The pair $(A\Delta , AL)$ is called the result of affine combinatorial patchworking. Glue by $s$ the sides of $A\Delta$. The resulting space $P\Delta$ is homeomorphic to the real projective plane $\mathbb{R}P^2$. Denote by $PL$ the image of $AL$ in $P\Delta$ and call the pair $(P\Delta ,PL)$ 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 $A\Delta \to\mathbb{R}^2$ mapping $AL$ onto the the affine curve defined by $b_t$ and a homeomorphism $P\Delta \to\mathbb{R}P^2$ mapping $PL$ onto the projective closure of this affine curve.

4.3 Examples of Patchworking

Patchworking of the Harnack curve of degree 6. Nine empty ovals and two nested ovals.

Patchworking of the Gudkov curve of degree 6. Five empty ovals and an oval enclosing five other empty ovals.

Patchworking of a counter-example to the Ragsdale Conjecture. A curve of degree 10 with 32 odd ovals constructed by Itenberg [2].

4.4 The Simplest Patchworking Coincides With Construction of Section 3.4

The polynomial $b_t$ defined by (5) is presented as $b_t^+-b_t^-$, where

$$b_t^{\varepsilon }(x,y)=\sum_{\text{\scriptsize $\begin{aligned}&... ...text{ at which $\sigma _{k,l}=\varepsilon $ }\end{aligned}$}}t^{\nu(k,l)}x^ky^l$$

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

$$\begin{multline*} p_h(x,y)=\sum_{k,l}a_{k,l}^{1/h}x^ky^l=\sum_{k,l}e^{-\nu(k,l)/... ... \sum_{k,l}e^{\nu(k,l)\ln t}x^ky^l=\sum_{k,l}t^{\nu(k,l)}x^ky^l. \end{multline*}$$

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

$$w=\max\{ku+lv-\nu(k,l)\;\mid\; (k,l) \tau \}.$$ (6)

In (6) we recognize the convex function conjugate to $\nu$. The graph of (6) is a convex PL surface, whose natural stratification is dual to the triangulation $\tau$ of $\Delta$: the face which lies on the plane $w=ku+lv-\nu(k,l)$ corresponds to the vertex $(k,l)$ of $\tau$, two such faces meet at an edge in the graph of (6) iff the corresponding vertices are connected with an edge of $\tau$, three faces meet at a vertex iff the corresponding vertices of $\tau$ belong to a triangle of $\tau$. In particular, we see that the configuration of planes satisfies the genericity condition of Section 3.4 and planes $w=ku+lv-\nu(k,l)$ corresponding to all monomials of $b_t^{\pm}$ show up in the graph of (6) as its faces.

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 $\Delta$.

Conclusion

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].