A new affine \(M\)-sextic. II. (English. Russian original) Zbl 0933.14039
Russ. Math. Surv. 53, No. 5, 1099-1101 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 243-244 (1998).
[For part I see Funct. Anal. Appl. 32, No. 2, 141-143 (1998); translation from Funkts. Anal. Prilozh, 32, No. 2, 91-94 (1998; Zbl 0932.14035).]
From the text: An affine real-valued algebraic curve is called an affine \(M\)-curve if it has the maximum possible number of connected components allowed by the Harnack inequality. In particular, an affine \(M\)-sextic has 16 connected components, 10 of which are ovals and the remaining 6 can be obtained from the 11th projectivisation oval by removing infinitely distant points. In the present note we extend the isotopic classification of affine \(M\)-sextics and present the details of a realization of the isotopy type \(A_3(0,5,5)\).
The construction is similar to the realization of \(B_2(1,8,1)\) presented in part I of this paper (loc. cit.). However, here we use a geometric argument, which makes it possible to shorten computations. The same argument can be applied to simplify the proof in part I.
From the text: An affine real-valued algebraic curve is called an affine \(M\)-curve if it has the maximum possible number of connected components allowed by the Harnack inequality. In particular, an affine \(M\)-sextic has 16 connected components, 10 of which are ovals and the remaining 6 can be obtained from the 11th projectivisation oval by removing infinitely distant points. In the present note we extend the isotopic classification of affine \(M\)-sextics and present the details of a realization of the isotopy type \(A_3(0,5,5)\).
The construction is similar to the realization of \(B_2(1,8,1)\) presented in part I of this paper (loc. cit.). However, here we use a geometric argument, which makes it possible to shorten computations. The same argument can be applied to simplify the proof in part I.
