Counter-Examples to Ragsdale Conjecture

The following theorem gives counter-examples to the Ragsdale Conjecture (or the conjecture of Petrovsky) (cf. [8]).

Theorem 8   For each integer $k \geq 1$

a) there exists a nonsingular real algebraic plane projective curve of degree $2k$ with

$$p = \frac{3k(k-1)}{2} + 1+ \left[\frac{(k - 3)^2 + 4}{8} \right]$$

b) there exists a nonsingular real algebraic plane projective curve of degree $2k$ with

$$n = \frac{3k(k-1)}{2} + \left[\frac{(k - 3)^2 + 4}{8} \right]$$

Remarks

The term $\left[\frac{(k - 3)^2 + 4}{8} \right]$ is positive when $k \geq 5$ (i. e., the first counter-examples to the Ragsdale Conjecture appear in degree 10; the patchwork of them is shown in Figures 2 and 3). On the other hand, it is known that there is no counter-example to the Ragsdale Conjecture among curves of lower degree. These counter-examples of degree 10 have the real schemes shown in Figure 8. One of these counter-examples can be improved in order to obtain a curve of degree 10 with $n = 32$. The corresponding patchworks (for $p = 32$ and $n = 32$) are shown in Figures 2 and 3.

Figure 8: Schemes of counter-examples to Ragsdale Conjecture of degree 10

Recently B. Haas [4] improved the construction presented below and obtained T-curves of degree $2k$ with

$$p=\frac{3k(k-1)}{2}+1+\left[\frac{k^2-7k+16}{6}\right].$$

Neither the counter-examples provided by the above theorem, nor curves constructed by Haas are M-curves. Moreover, as we mentioned above, it is not known if the conjecture of Petrovsky holds for M-curves. The only known counter-examples to the Ragsdale Conjecture among M-curves are the curves constructed by Viro [16] (see also [18]). It is curious that we did not succeed in presenting those M-counter-examples as T-curves.

Proof. Proof of Theorem Let us show, first, how to construct a curve of degree $m = 2k$ with $p =\frac {3k(k-1)}{2} + 2$.

Suppose that the hexagon $S$ shown in Figure 9 is placed inside of the triangle $T$ in such a way that the center of $S$ has both coordinates odd. Any convex primitive triangulation of a convex part of a convex polygon is extendable to a convex primitive triangulation of the polygon. Inside of the hexagon $S$, let us take the convex primitive triangulation shown in Figure 9 and extend it to $T$.

Figure 9: Hexagon $S$ and the patchwork fragment produced by it.

To apply the Patchwork Theorem we need to choose signs at the vertices. Inside of $S$ put signs according to Figure 9; outside use the Harnack rule of distribution of signs.

It is easy to calculate that the corresponding piecewise-linear curve $\mathcal{L}$ has exactly one even oval more than the Harnack curve constructed above (i. e. now $p =\frac {3k(k-1)}{2} + 2$). One can verify that the curve obtained has the real scheme shown in Figure 10.

Figure 10: Real scheme of a curve of degree $2k$ with $p =\frac {3k(k-1)}{2} + 2$.

Consider the partition of the triangle $T$ shown in Figure 11. Let us take in each shadowed hexagon the triangulation and the signs of the hexagon $S$. The triangulation of the union of the shadowed hexagons can be extended to the primitive convex triangulation of $T$. Let us fix such an extension. Outside of the union of the shadowed hexagons choose the signs at the vertices of the triangulation using the Harnack rule.

Figure 11: Partition of $T$ for part a.

Calculation shows that for the corresponding piecewise-linear curve $\mathcal{L}$

$$p = \frac{3k(k-1)}{2} +1+ a$$

where $a$ is the number of shadowed hexagons, and

$$a = \left[\frac{(k - 3)^2 + 4}{8} \right]$$

This curve has the real scheme shown in Figure 12.

Figure 12: Real scheme of a curve of degree $2k$ with $p = \frac {3k(k-1)}{2} +1+ a$.

To prove part b) of the theorem, let us take again the partition of the triangle $T$ shown in Figure 11 with the triangulation and the signs of each shadowed hexagon coinciding with the triangulation and the signs of $S$. Fix, in addition, a triangulation of a neighborhood of the axis $OY$ and the signs at the vertices of the triangulation as shown in Figure 13 (the case $k \equiv 1(\mod4)$). The chosen triangulation of the union of the shadowed hexagons and the neighborhood of the $y$-axis can be extended to a primitive convex triangulation of $T$. Outside of the union of the shadowed hexagons and the neighborhood of the $y$-axis let us again choose the signs at the vertices of the triangulation using the Harnack rule.

Figure 13: Partition of $T$ for part b.

The corresponding piecewise-linear curve $\mathcal{L}$ has the real scheme shown in Figure 14. In this case $n=\frac {3k(k-1)}{2}+a$. $\qedsymbol$

Figure 14: Real scheme of a curve of degree $2k$ with $n=\frac {3k(k-1)}{2}+a$.