Вычислительная математика - ИСЭМ СО РАН

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ON SIGNDEFINITENESS OF THE FORMS OF TWO FORTH-ORDER
VARIABLES
M.A. Novickov
Institute for System Dynamics and Control Theory, Siberian Branch, Russian Academy of
Sciences
e-mail: nma@icc.ru
Abstract. In the present paper, via an example of a forth-order form for two variables we propose a
new method of solving the problem of signdefiniteness for homogeneous forms. The method implies
the power-oriented replacement of the variables by which it is possible to conduct reduction to the
quadratic form. As a result of such a transformation we obtain another quadratic form which is equal
to zero. The linear bundle of quadratic forms obtained gives a parametrized quadratic form. Afater
reducing the latter to the sum of full squares, the signdefiniteness of the quadratic parametric matrix
is reduced to a conventionally extremum problem with only one parameter. The requirement of
positiveness of coefficients with the full squares leads to obtaining necessary and sufficient conditions
of signdefiniteness of the parametric quadratic form. Complete correspondence of signdefiniteness,
signvariability and signconstancy of the parametric quadratic and initial forms has been identified.
Keywords: form, quadratic form, signdefinite form, signvariable form, signconstant form, extremum
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