Combinatorial Look on Patchworking

Initial Data. Let $m$ be a positive integer (it will be the degree of the curve under construction) and $T$ be the triangle in $\mathbb{R}^2$ with vertices $(0,0)$, $(m,0)$, $(0,m)$. Let $\tau$ be a triangulation of $T$ with vertices having integer coordinates and equipped with signs. The sign (plus or minus) at the vertex with coordinates $(i,j)$ is denoted by $\sigma _{i,j}$.

Construction of Piecewise Linear Curve. Take copies

$$T_{x} = s_x(T),\;\; T_{y} = s_y(T),\;\; T_{xy} = s(T)$$

of $T$, where $s= s_x \circ s_y$ and $s_x,\; s_y$ are reflections with respect to the coordinate axes. Denote by $T_*$ the square $T \cup T_x \cup T_y \cup T_{xy}$. Extend the triangulation $\tau$ to a symmetric triangulation of $T_*$, 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 $T_*$ has vertices of different signs, select a midline separating pluses from minuses. Denote by $L$ the union of the selected midlines. It is a collection of polygonal lines contained in $T_*$. The pair $(T_*, L)$ is called the result of affine combinatorial patchworking. Glue by $s$ the sides of $T_*$. The resulting space $\mathcal{T}$ is homeomorphic to the real projective plane $\mathbb{R}P^2$. Denote by $\mathcal{L}$ the image of $L$ in $\mathcal{T}$ and call the pair $(\mathcal{T},\mathcal{L})$ the result of projective combinatorial patchworking.

Figure 1: Initial data and the result of combinatorial patchworking of it.

Let us introduce an additional assumption: the triangulation $\tau$ of $T$ is convex. This means that there exists a convex piecewise-linear function $T \longrightarrow {\mathbb{R}}$ which is linear on each triangle of $\tau$ and not linear on the union of any two triangles of $\tau$.

1 (Patchwork Theorem)   Under the assumptions above on the triangulation $\tau$ of $T$, there exist a nonsingular real algebraic plane affine curve of degree $m$ and a homeomorphism of the plane $\mathbb{R}^2$ onto the interior of the square $T_*$ mapping the set of real points of this curve onto $L$. Furthermore, there exists a homeomorphism $\mathbb{R}P^2\to\mathcal{T}$ mapping the set of real points of the corresponding projective curve onto $\mathcal{L}$.

Figure 2: Patchwork of a counter-example to Ragsdale Conjecture with degree 10 and p=32.

Figure 3: Patchwork of a counter-example to Ragsdale Conjecture with degree 10 and n=32.