D. A. Gudkov

ON THE ARRANGEMENT OF THE OVALS OF A CURVE OF THE SIXTH ORDER

(Presented by Academician I. G. Petrovskii, 5 VII 1968)

MATHEMATICS

1. A nonsingular plane real curve of the 6th order \(C_6\)* can have, by Harnack \((^1)\), at most 11 ovals. We shall say that a curve \(C_6\) has type \(\dfrac{k}{1}\,l\) (or is a \(C_6\)-curve \(\dfrac{k}{1}\,l\)) if it has a certain oval \(\alpha\) (the principal one), inside which there lie \(k\) ovals exterior to one another and \(l\) ovals outside the oval \(\alpha\) and exterior to one another (see \((^{11})\)). The type of a curve \(C_6\) consisting of \(k\) ovals exterior to one another will be denoted by \(k\). A curve \(C_6\) can have the following logically possible types with 11, 10, and 9 ovals:

Table 1

\[ \begin{gathered} \dfrac{10}{1};\quad \dfrac{9}{1}1;\quad \dfrac{8}{1}2;\quad \dfrac{7}{1}3;\quad \dfrac{6}{1}4;\quad \dfrac{5}{1}5;\quad \dfrac{4}{1}6;\quad \dfrac{3}{1}7;\quad \dfrac{2}{1}8;\quad \dfrac{1}{1}9;\quad 11,\\[4pt] \dfrac{9}{1};\quad \dfrac{8}{1}1;\quad \dfrac{7}{1}2;\quad \dfrac{6}{1}3;\quad \dfrac{5}{1}4;\quad \dfrac{4}{1}5;\quad \dfrac{3}{1}6;\quad \dfrac{2}{1}7;\quad \dfrac{1}{1}8;\quad 10,\\[4pt] \dfrac{8}{1},\quad \dfrac{7}{1}1;\quad \dfrac{6}{1}2;\quad \dfrac{5}{1}3;\quad \dfrac{4}{1}4;\quad \dfrac{3}{1}5;\quad \dfrac{2}{1}6;\quad \dfrac{1}{1}7;\quad 9 \end{gathered} \]

A. Harnack constructed a curve \(C_6\) of type \(\dfrac{1}{1}9\) \((^1)\), D. Hilbert one of type \(\dfrac{9}{1}1\) \((^2)\). D. Hilbert, in formulating his 16th problem \((^3)\), expressed the conviction that from the first row of Table 1 there exist only curves of the types \(\dfrac{9}{1}1\) and \(\dfrac{1}{1}9\). H. Kahn \((^4)\) and K. Lebenstein \((^5)\) tried to prove (without success) that a curve \(C_6\) of type 11 does not exist. K. Rohn tried to prove the nonexistence of curves \(C_6\) of types \(\dfrac{10}{1}\) and \(11\) \((^6)\), but there are errors in his work. The nonexistence of a curve \(C_6\) of type 11 follows from the theorem of I. G. Petrovskii \((^9)\). We asserted \((^{11})\) that a curve cannot have the types situated above the broken line in Table 1, and also the types \(\dfrac{5}{1}3\), \(\dfrac{5}{1}4\), and \(\dfrac{5}{1}5\). However, in the proofs (which were not published) of the nonexistence of curves \(C_6\) of types \(\dfrac{5}{1}3\), \(\dfrac{5}{1}4\), \(\dfrac{5}{1}5\), \(\dfrac{6}{1}3\), \(\dfrac{6}{1}4\), we made errors. After correcting the errors and carrying out additional analysis it turned out that curves \(\dfrac{6}{1}3\) and \(\dfrac{6}{1}4\) do not exist, the curve \(\dfrac{5}{1}3\) is constructed by D. Hilbert’s method, and the curves \(\dfrac{5}{1}4\) and \(\dfrac{5}{1}5\) exist, but are not constructed by the methods of A. Harnack and D. Hilbert.

Thus, curves \(C_6\) of the types situated below the broken line in Table 1 exist, while those above the broken line do not exist. (For complete proofs of this assertion see \((^{13})\).)

* By \(C_m\), \(\overline{C}_m\), etc., in this note is denoted a plane real curve of the \(m\)-th order.

1. In the present note we shall set forth the idea of constructing a curve of type \(\dfrac{5}{1}\,5\) (and consequently also \(\dfrac{5}{1}\,4\)).

Lemma 1. Let \(F(x)\) be an irreducible real simple (i.e., all its singular points are simple double points) curve of degree 6, let \(y_0, y_1, y_2\) be singular points of the curve \(F\), and let the lines \(x_i \equiv y_k y_l\) \((i,k,l=0,1,2\) and distinct) not be tangent to the curve \(F\). Then, under the quadratic transformation

\[ x_0=y_1y_2,\qquad x_1=y_0y_2,\qquad x_2=y_0y_1 \tag{1} \]

(with fundamental points \(y_0, y_1, y_2\) and fundamental lines \(x_0, x_1, x_2\)), the curve \(F(x)\) is transformed in the plane \((y)\) into a curve \(F_{(1)}(y)\)—an irreducible real simple curve of degree 6, having at the points \(x_0, x_1, x_2\) simple double points. The remaining singular points of the curve \(F(x)\) (except for \(y_0, y_1, y_2\)) are transformed into the same kind of singular points of the curve \(F_{(1)}(y)\), and the transformation (1) establishes a one-to-one and continuous correspondence between the branches of the curves \(F\) and \(F_{(1)}\).

The proof follows easily from the theorems of § 7, Chapter 3 \((^{10})\).

Lemma 2. Let \(F\) be a simple real unicursal curve of degree 6 and let all its (10) singular points be real.

Then, by an arbitrarily small change of the coefficients of the curve \(F\), one can obtain a curve \(\Phi\) of degree 6 isotopic to \(F\), for which the line through any two singular points is not tangent to the curve \(\Phi\).

The proof is not difficult.

Theorem 1. There exists a curve \(C_6\) of type \(\dfrac{5}{1}\,5\).

Proof. Let \(C_3\) be a unicursal curve with a node \(z_1\) (see Fig. 1a). On \(C_3\) there exists a point \(z_2\) such that on the arc \(z_1z_2\) there are no inflection points of the curve \(C_3\). By rotating the line \(z_1z_2\) about the point \(z_2\), we obtain a line \(L_1\), intersecting \(C_3\) further at the points \(z_3\) and \(z_4\) on the loop of the curve \(C_3\). The tangent to \(C_3\) at the point \(z_2\) intersects \(C_3\) further at the point \(Q\). We may assume that \(z_2\) is so close to \(z_1\) that the whole portion of the curve \(C_3\) \(Qz_1z_2\) lies in the finite part of the plane. Next, by rotating the line \(z_1z_2\) about the point \(O\) (lying on the exterior segment \(z_1z_2\)), we obtain a line \(L_2\), intersecting the tangent \(Qz_2\) at a point \(P\), lying on the finite segment \(Qz_2\), and \(C_3\) at the points \(z_6, z_7\) (on the arc \(z_1z_2\)) and \(z_5\). \(L_2\) intersects \(L_1\) at the point \(z_8\). Finally, let us move the tangent \(z_2Q\) into the position \(L_3\) in the following way: move the point of tangency to \(z_9\) (in the direction from \(z_7\) to \(z_2\)), then rotate the tangent to \(C_3\) at \(z_9\) about \(z_9\) so as to obtain a line intersecting \(C_3\) at a point \(z_{10}\), close to \(z_9\) and situated on the other side of \(z_9\) than the point \(z_2\), and also at some point \(z_{11}\) (close to \(Q\)). \(L_3\) intersects the lines \(L_1\) and \(L_2\) at the points \(z_{12}\) and \(z_{13}\). Observe that, under the displacement of \(Qz_2\) into \(L_3\), the point \(P\) passes into \(z_{13}\). This displacement may be assumed so small that the line \(z_2z_{13}\) has with \(C_3\), on the arc \(z_1Oz_{11}\), two points of intersection. The curve \(C_3L_1L_2L_3 \equiv \Phi\) is a simple reducible curve of degree 6 with 13 nodes. By Theorem 10, § 6 \((^{12})\), one can bifurcate the nodes \(z_8, z_4\), and \(z_{10}\) and preserve the remaining nodes of the curve \(\Phi\), so as to obtain a simple unicursal curve \(F\) of Fig. 1b. Choose the fundamental points \(y_0, y_1, y_2\) of the transformation (1) at \(z_2, z_{13}\), and \(z_{12}\). In the plane \((y)\) we obtain (by Lemmas 1 and 2) a curve \(F_{(1)}(y)\) with new nodes \(x_0, x_1, x_2\). The nodes \(z_2, z_{13}\), and \(z_{12}\) disappear, while the remaining nodes of the curve \(F(x)\) are preserved, and we denote them by the former letters. Then subject the curve \(F_{(1)}(y)\) to the transformation (1), choosing as fundamental points \(y_0', y_1', y_2'\) the points \(z_5, z_1\), and \(z_3\). Let \(x_0', x_1', x_2'\) be the corresponding fundamental lines. Then in the plane \((y')\) we obtain a curve \(F_{(2)}(y')\) with simple double points \(x_0', x_1', x_2'; x_0, x_1, x_2; z_6, z_9, z_{11}\). Choose the fundamental points \(y_0'', y_1'', y_2''\) of the transformation (1) of the curve \(F_{(2)}(y')\) at the points \(x_0, z_9, z_{11}\), and denote the fundamental lines by \(x_0'', x_1'', x_2''\). In the plane \((y'')\), we obtain a curve \(F_{(3)}(y'')\) with singular points \(x_0'', x_1'', x_2''; x_0'\),

\(x_1',\ x_2';\ x_1,\ x_2,\ z_6,\ z_7\). Finally, for the curve \(F_{(3)}(y''')\) we choose the fundamental points \(y_0''',\ y_1''',\ y_2'''\) of the transformation (1) at the points \(x_0',\ z_6,\ z_7\), and denote the fundamental lines through them by \(x_0''',\ x_1''',\ x_2'''\). In the plane \((y''')\) we obtain the curve \(F_{(4)}(y''')\)—a simple unicursal curve of the 6th order (Fig. 1в) of type \(\dfrac{5}{1}\,5\). Under the last three quadratic transformations the lines \(x_1', x_2';\ x_0'', x_1'', x_2''\) and \(x_0''', x_1''', x_2'''\) may intersect the corresponding curves \(F_{(1)}, F_{(2)}\), and \(F_{(3)}\) in two real points. Then the corresponding points of the curve in Fig. 1в will be nodes analogous to the nodes \(x_1, x_2\). The curve \(F_{(4)}\) in this case always has type \(\dfrac{5}{1}\,5\). The theorem is proved.

Fig. 1

Theorems 2 and 3 can be proved.

Theorem 2.
\(1^\circ\). A curve \(C_6\) of type \(\dfrac{9}{1}\,1\) can be obtained from a simple reducible curve only of type \(C_2 \cdot C_4\).

\(2^\circ\). A curve \(C_6\) of type \(\dfrac{5}{1}\,5\) can be obtained from a simple reducible curve only of type \(C_1 \cdot C_5\) (Fig. 1a).

\(3^\circ\). A curve \(C_6\) of type \(\dfrac{1}{1}\,9\) can be obtained from simple reducible curves of each of the types \(C_2 \cdot C_4;\ C_1 \cdot C_5;\ C_3 \cdot C_3\).

A. Wiman \((^{7})\) showed that there exist curves of even orders \(m \geqslant 8\) with the greatest number of ovals (\(M\)-curves) which are not constructed by the methods of Harnack and Hilbert (see also \((^{8})\), Chapter 4, § 2).

Theorem 3. From the curve \(C_6 = C_1 \cdot C_5\) of Fig. 1, by applying Harnack’s construction, one obtains a series of \(M\)-curves. In particular, the following curves are obtained:

1) For odd \(m \geqslant 7\), a curve of type
\[ \mathrm{I}\,\frac{5}{1}\left\{\frac{m^2-3m}{2}-5\right\}, \]
i.e., having an odd branch \(\mathrm{I}\), five ovals inside one of them, and \(\frac{m^2-3m}{2}-5\) exterior ovals (outside one another).

2) For even \(m \geqslant 6\), a curve of type
\[ \frac{5}{1}\,\frac{\left\{\frac18 m(m-6)\right\}}{1} \left\{\frac{3m^2-6m}{8}-4\right\}, \]
which are not constructed by the methods of A. Harnack, D. Hilbert, and A. Wiman \((^{1,2,7})\).

Gorky State University
named after N. I. Lobachevsky

Received
2 VII 1968

CITED LITERATURE

\({}^{1}\) H. Hornack, Math. Ann., 10, 189 (1876).
\({}^{2}\) D. Hilbert, Math. Ann., 38, 115 (1891).
\({}^{3}\) D. Hilbert, Arch. Math. u. Phys., 3 Reihe, 1, 44 (1901).
\({}^{4}\) G. Kahn, Inaugural Dissertation, Göttingen, 43 pp., 1909.
\({}^{5}\) K. Löbenstein, Inaugural Dissertation, Göttingen, 37 pp., 1910.
\({}^{6}\) K. Rohn, Math. Ann., 73, 177 (1913).
\({}^{7}\) A. Wiman, Math. Ann., 90, 222 (1923).
\({}^{8}\) I. L. Coolidge, A Treatise on Algebraic Plane Curves, Oxford, 1931.
\({}^{9}\) I. Petrovsky, Ann. Math., 39, No. 1, 187 (1938).
\({}^{10}\) R. Walker, Algebraic Curves, Moscow, 1952.
\({}^{11}\) D. A. Gudkov, DAN, 48, No. 4, 521 (1954).
\({}^{12}\) D. A. Gudkov, Matem. sborn., 67 (109), 4, 481 (1965).
\({}^{13}\) D. A. Gudkov, Uch. zap. Gorky Univ., issue 87 (1968).