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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 1, Pages 71–77 (Mi vuu466)  

This article is cited in 3 scientific papers (total in 3 papers)

MATHEMATICS

Cubic forms without monomials in two variables

A. V. Seliverstov

Laboratory 6, Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Bol'shoi Karetnyi per., 19, build. 1, Moscow, 127051, Russia

Abstract: It is proved that a general cubic form over the field of complex numbers can be transformed into a form without monomials of exactly two variables by means of a non-degenerate linear transformation of coordinates. If the coefficients of monomials in only one variable are equal to one, and the remaining coefficients belong to sufficiently small polydisc near zero, then the transformation can be approximated by iterative algorithm. Under these restrictions the same result holds over the reals. This result generalizes the Levy–Desplanques theorem on strictly diagonally dominant matrices. We discuss in detail the properties of reducible cubic forms. So we prove the existence of a reducible real cubic form that is not equivalent to any form with all monomials in only one variable and without any monomials in exactly two variables. We suggest a sufficient condition for the existence of a singular point on a projective cubic hypersurface. The computational complexity of singular points recognition is discussed.

Keywords: cubic form, linear transformation, singular point.

Funding Agency Grant Number
Russian Foundation for Basic Research 13--04--40196--Н КОМФИ


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Document Type: Article
UDC: 512.647
MSC: 15A69, 14J70, 32S25
Received: 16.01.2015

Citation: A. V. Seliverstov, “Cubic forms without monomials in two variables”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:1 (2015), 71–77

Citation in format AMSBIB
\Bibitem{Sel15}
\by A.~V.~Seliverstov
\paper Cubic forms without monomials in two variables
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2015
\vol 25
\issue 1
\pages 71--77
\mathnet{http://mi.mathnet.ru/vuu466}
\elib{http://elibrary.ru/item.asp?id=23142053}


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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. A. V. Seliverstov, “O simmetrii proektivnykh krivykh”, Vestnik TvGU. Seriya: Prikladnaya matematika, 2016, no. 3, 59–66  mathnet  crossref  elib
    2. V. A. Lyubetsky, A. V. Seliverstov, “A novel algorithm for solution of a combinatory set partitioning problem”, J. Commun. Technol. Electron., 61:6 (2016), 705–708  crossref  isi  scopus
    3. A. V. Seliverstov, “O kasatelnykh pryamykh k affinnym giperpoverkhnostyam”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 248–256  mathnet  crossref  elib
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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