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Mat. Zametki, 1997, Volume 62, Issue 6, Pages 831–835 (Mi mz1672)  

This article is cited in 1 scientific paper (total in 1 paper)

The number of components of complements to level surfaces of partially harmonic polynomials

V. N. Karpushkin

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: In this paper $k$-harmonic polynomials in $\mathbb R^n$ i.e. polynomials satisfying the Laplace equation with respect to $k$ variables: $(\partial^2/\partial x_1^2+…+\partial^2/\partial x_k^2)F=0$ are considered; here $1\le k\le n$, $n\ge2$. For a polynomial $F$ (of degree $m$) of this type, it is proved that the number of components of the complements of its level sets does not exceed $2m^{n-1}+O(m^{n-2})$. Under the assumptions that the singular set of the level surface is compact or that the leading homogeneous part of the $k$-harmonic polynomial $F$ is nondegenerate, sharper estimates are also established.

DOI: https://doi.org/10.4213/mzm1672

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English version:
Mathematical Notes, 1997, 62:6, 697–700

Bibliographic databases:

UDC: 513.62
Received: 22.09.1995
Revised: 15.05.1997

Citation: V. N. Karpushkin, “The number of components of complements to level surfaces of partially harmonic polynomials”, Mat. Zametki, 62:6 (1997), 831–835; Math. Notes, 62:6 (1997), 697–700

Citation in format AMSBIB
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\by V.~N.~Karpushkin
\paper The number of components of complements to level surfaces of partially harmonic polynomials
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\yr 1997
\vol 62
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\zmath{https://zbmath.org/?q=an:1033.14035}
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\transl
\jour Math. Notes
\yr 1997
\vol 62
\issue 6
\pages 697--700
\crossref{https://doi.org/10.1007/BF02355457}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. N. Karpushkin, “The number of connected components of a level surface of a harmonic polynomial”, Russian Math. Surveys, 65:4 (2010), 783–784  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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