3.3 A Survey of Prohibitions on the Real Schemes Which Come from Topology

In this section I list all prohibitions on the real scheme of a flexible curve of degree $m$ that I am aware of, including the ones already referred to above, but excluding prohibitions which follow from the other prohibitions given here or from the prohibitions on the complex schemes which are given in the next section.

3.3.A   A curve is one-sided if and only if it has odd degree.

This fact was given before as a corollary of Bézout's theorem (see Section 1.3) and proved for flexible curves in Section 3.2 (Theorem 3.2.G).

3.3.B. Harnack's Inequality.   The number of components of the set of real points of a curve of degree $m$ is at most $\frac{(m-1)(m-2)}2+1$.

Harnack's inequality is undoubtedly the best known and most important prohibition. It can also be deduced from Bézout's theorem (cf. Section 1.3) and was proved for flexible curves in Section 3.2 (Theorem 3.2.A).

In prohibitions 3.3.C-3.3.P the degree $m$ of the curve is even: $m=2k$.

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Extremal Properties of Harnack's Inequality

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3.3.C. Gudkov-Rokhlin Congruence.   In the case of an M-curve (i.e., if $p+n=(m-1)(m-2)/2+1$),

$$p-n\equiv k^2\quad \mod 8.$$

3.3.D. Gudkov-Krakhnov-Kharlamov Congruence.   In the case of an
$(M-1)$-curve (i.e., if $p+n=\frac{(m-1)(m-2)}2$),

$$p-n\equiv k^2\pm 1\quad\mod 8.$$

The Euler characteristic of a component of the complement of a curve in $\mathbb{R}P^2$ is called the characteristic of the oval which bounds the component from outside. An oval with a positive characteristic is said to be elliptic, an oval with the zero characteristic is said to be parabolic and an oval with a negative characteristic is said to be hyperbolic.

3.3.E. Fiedler's Congruence.   If the curve is an M-curve, $m\equiv 4\mod 8$, and every even oval has an even characteristic, then

$$p-n\equiv -4\quad\mod 16.$$

3.3.F. Nikulin's Congruence.   If the curve is an M-curve, $m\equiv 0\mod 8$, and the characteristic of every even oval is divisible by $2^r$, then

$$p-n\equiv 0\mod 2^{r+3},$$ (3)

$$p-n= 4^q\chi,$$ (4)

where $q\ge 2$ and $\chi\equiv 1\mod 2$.

3.3.G. Nikulin's Congruence.   If the curve is an M-curve, $m\equiv 2\mod 4$ and the characteristic of every odd oval is divisible by $2^r$, then

$$p-n\equiv 1\quad\mod 2^{r+3}.$$

Denote the number of even ovals with positive characteristic by $p^+$, the number of even ovals with zero characteristic by $p^0$, and the number of even ovals with negative characteristic by $p^-$. Similarly define $n^+,n^0$ and $n^-$ for the odd ovals; and let $l^+,l^0$ and $l^-$ be the corresponding numbers for both even and odd ovals together.

Refined Petrovsky Inequalities

3.3.H   $p-n^-\le \frac{3k(k-1)}2+1$.

3.3.I   $n-p^-\le \frac{3k(k-1)}2$.

Refined Arnold Inequalities

3.3.J   $p^-+p^0\le\frac{(k-1)(k-2)}2+\frac{1+(-1)^k}2$.

3.3.K   $n^-+n^0\le \frac{(k-1)(k-2)}2$.

Extremal Properties of the Refined Arnold Inequalities

3.3.L   If $k$ is even and $p^-+p^0=\frac{(k-1)(k-2)}2+1$, then $p^-=p^+=0$.

3.3.M   If $k$ is odd and $n^-+n^0=\frac{(k-1)(k-2)}2$, then $n^-=n^+ =0$ and there is only one outer oval at all.

Viro-Zvonilov Inequalities

Besides Harnack's inequality, we know only one family of prohibition coming from topology which extends to real schemes of both even and odd degree. For proofs see [VZ-92].

3.3.N. Bound of the Number of Hyperbolic Ovals.   The number of components of the complement of a curve of odd degree $m$ that have a negative Euler characteristic does not exceed $\frac{(m-3)^2}4$. In particular, for any odd $m$

$$l^-\le \frac{(m-3)^2}4.$$

The latter inequality also holds true for even $m\ne 4$, but it follows from Arnold inequalities 3.3.J and 3.3.K.

3.3.O. Bound of the Number of Nonempty Ovals.   If $h$ is a divisor of $m$ and a power of an odd prime, and if $m\ne 4$, then

$$l^-+l^0\le \frac{(m-3)^2}4+\frac{m^2-h^2}{4h^2}.$$

If $m$ is even, this inequality follows from 3.3.J-3.3.L.

3.3.P. Extremal Property of the Viro-Zvonilov Inequality.   If

$$l^-+l^0=\frac{(m-3)^2}4+\frac{m^2-h^2}{4h^2},$$

where $h$ is a divisor of $m$ and a power of an odd prime $p$, then there exist $\alpha_1,\dotsc,\alpha_r\in \mathbb{Z}_p$ and components $B_1,\dotsc,B_r$ of the complement $\mathbb{R}P^2\setminus \mathbb{R}A$ with $\chi(B_1)=\cdots=\chi(B_r)=0$, such that the boundary of the chain $\sum^r_{i=1}\alpha_i[B_i]\in C_2(\mathbb{R}P^2;\mathbb{Z}_p)$ is $[\mathbb{R}A]\in C_1(\mathbb{R}P^2;\mathbb{Z}_p)$.