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Russia Moscow
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 Sciences^{a} 
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 nondegenerate 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 LevyDesplanques 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. 7177. 
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