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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
##### Keywords:
components; level surface; harmonic polynomial
Full Text:
##### References:
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