RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (166 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 513.62
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
\Bibitem{Kar97} \by V.~N.~Karpushkin \paper The number of components of complements to level surfaces of partially harmonic polynomials \jour Mat. Zametki \yr 1997 \vol 62 \issue 6 \pages 831--835 \mathnet{http://mi.mathnet.ru/mz1672} \crossref{https://doi.org/10.4213/mzm1672} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1635154} \zmath{https://zbmath.org/?q=an:1033.14035} \elib{http://elibrary.ru/item.asp?id=13250234} \transl \jour Math. Notes \yr 1997 \vol 62 \issue 6 \pages 697--700 \crossref{https://doi.org/10.1007/BF02355457} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000075396200023} 

• http://mi.mathnet.ru/eng/mz1672
• https://doi.org/10.4213/mzm1672
• http://mi.mathnet.ru/eng/mz/v62/i6/p831

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 187 Full text: 82 References: 32 First page: 2