1.11 Analysis of the Results of the Constructions. Ragsdale

In 1906, V. Ragsdale [Rag-06] made a remarkable attempt to guess new prohibitions, based on the results of the constructions by Harnack's and Hilbert's methods. She concentrated her attention on the case of curves of even degree, motivated by the following special properties of such curves. Since a curve of an even degree is two-sided, it divides $\mathbb{R}P^2$ into two parts, which have the curve as their common boundary. One of the parts contains a nonorientable component; it is denoted by $\mathbb{R}P^2_-$. The other part, which is orientable, is denoted by $\mathbb{R}P^2_+$. The ovals of a curve of even degree are divided into inner and outer ovals with respect to $\mathbb{R}P^2_+$ (i.e., into ovals which bound a component of $\mathbb{R}P^2_+$ from the inside and from the outside). Following Petrovsky [Pet-38], one says that the outer ovals with respect to $\mathbb{R}P^2_+$ are the even ovals (since such an oval lies inside an even number of other ovals), and the rest of the ovals are called odd ovals. The number of even ovals is denoted by $p$, and the number of odd ovals is denoted by $n$. These numbers contain very important information about the topology of the sets $\mathbb{R}P^2_+$ and $\mathbb{R}P^2_-$. Namely, the set $\mathbb{R}P^2_+$ has $p$ components, the set $\mathbb{R}P^2_-$ has $n+1$ components, and the Euler characteristics are given by $\chi(\mathbb{R}P^2_+ )=p-n$ and $\chi(\mathbb{R}P^2_-)=n-p+1$. Hence, one should pay special attention to the numbers $p$ and $n$. (It is amazing that essentially these considerations were stated in a paper in 1906!)

By analyzing the constructions, Ragsdale [Rag-06] made the following observations.

1.11.A (compare with 1.9.A and 1.9.B)   For any Harnack M-curve of even degree $m$,

$$p=(3m^2-6m+8)/8,\qquad n=(m^2-6m+8)/8.$$

1.11.B   For any Hilbert M-curve of even degree $m$,

$$(m^2-6m+16)/8\le p\le (3m^2-6m+8)/8,$$

$$(m^2-6m+8)/8\le n\le (3m^2-6m)/8.$$

This gave her evidence for the following conjecture.

1.11.C Ragsdale Conjecture.   For any curve of even degree $m$,

$$p\le (3m^2-6m+8)/8,\qquad n\le (3m^2-6m)/8.$$

The most mysterious in this problem seems to be its number. The number sixteen plays a very special role in topology of real algebraic varieties. It is difficult to believe that Hilbert was aware of that. It became clear only in the beginning of seventies (see Rokhlin's paper ``Congruences modulo sixteen in the sixteenth Hilbert's problem'' [Rok-72]). Nonetheless, sixteen was the number assigned by Hilbert to the problem.

Writing cautiously, Ragsdale formulated also weaker conjectures. About thirty years later I. G. Petrovsky [Pet-33], [Pet-38] proved one of these weaker conjectures. See below Subsection 1.13.

Petrovsky also formulated conjectures about the upper bounds for $p$ and $n$. His conjecture about $n$ was more cautious (by 1).

Both Ragsdale Conjecture formulated above and its version stated by Petrovsky [Pet-38] are wrong. However they stayed for rather long time: Ragsdale Conjecture on $n$ was disproved by the author of this book [Vir-80] in 1979. However the disproof looked rather like improvement of the conjecture, since in the counterexamples $n=\frac{3k(k-1)}2+1$. Drastically Ragsdale-Petrovsky bounds were disproved by I. V. Itenberg [Ite-93] in 1993: in Itenberg's counterexamples the difference between $p$ (or $n$) and $\frac{3k(k-1)}2+1$ is a quadratic function of $k$.

The numbers $p$ and $n$ introduced by Ragsdale occur in many of of the prohibitions that were subsequently discovered. While giving full credit to Ragsdale for her insight, we must also say that, if she had looked more carefully at the experimental data available to her, she should have been able to find some of these prohibitions. For example, it is not clear what stopped her from making the conjecture which was made by Gudkov [GU-69] in the late 1960's. In particular, the experimental data could suggest the formulation of the Gudkov-Rokhlin congruence proved in [Rok-72]: for any M-curve of even degree $m=2k$

$$p-n\equiv k^2\mod8$$

Maybe mathematicians trying to conjecture restrictions on some integer should keep this case in mind as an evidence that restrictions can have not only the shape of inequality, but congruence. Proof of these Gudkov's conjectures initiated by Arnold [Arn-71] and completed by Rokhlin [Rok-72], Kharlamov [Kha-73], Gudkov and Krakhnov [GK-73] had marked the beginning of the most recent stage in the development of the topology of real algebraic curves. We shall come to this story at the end of this Section.