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The number of components of the complement of the level surface of a harmonic polynomial in three variables. (English. Russian original) Zbl 0846.57028
Funct. Anal. Appl. 28, No. 2, 116-118 (1994); translation from Funkts. Anal. Prilozh. 28, No. 2, 52-54 (1994).
We establish an upper bound for the number of components of the complement of the level surface \(W\) for a harmonic polynomial \(F\) of degree \(n\) in three variables in \(\mathbb{R}^3\), namely, \(n^2 - n + 2\), and also for the number of components of the complement of the closure \(\overline W \subset \mathbb{R} P^3\) of this surface; in the latter case the bound is \(n^2 - 2n + 4\) for even \(n\) and \(n^2 - 2n + 2\) for odd \(n\). General theorems of algebraic geometry imply upper bounds of the order of \(n^3\) for the number of components of the complement; see [D. A. Gudkov, Russ. Math. Surv. 29, No. 4, 1-79 (1974); translation from Usp. Mat. Nauk 29, No. 4(178), 3-79 (1974; Zbl 0316.14018)]. For some other theorems on level surfaces of harmonic polynomials see [the author, Vestn. Mosk. Univ., Ser. I 1981, No. 4, 3-4 (1981; Zbl 0482.14015); Funct. Anal. Appl. 19, 295-299 (1985); translation from Funkts. Anal. Prilozh. 19, No. 4, 55-60 (1985; Zbl 0606.14020)]; ibid. 23, No. 3, 218-220 (1989); translation from ibid. 23, No. 3, 59-60 (1989; Zbl 0701.47030); ibid. 26, No. 4, 300-302 (1992); translation from ibid. 26, No. 4, 86-88 (1992; Zbl 0799.14032)].

MSC:
57R99 Differential topology
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