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Home » Issues » Archive of Issues » 2015 - 1 issue » pp. 71-77

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Russia Moscow
Year
2015
Volume
25
Issue
1
Pages
71-77
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Section Mathematics
Title Cubic forms without monomials in two variables
Author(-s) Seliverstov A.V.a
Affiliations Institute for Information Transmission Problems, Russian Academy of Sciencesa
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
UDC 512.647
MSC 15A69, 14J70, 32S25
DOI 10.20537/vm150108
Received 16 January 2015
Language Russian
Citation Seliverstov A.V. Cubic forms without monomials in two variables, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 1, pp. 71-77.
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