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SM ISO690:2012 BUJAC, Cristina. One subfamily of cubic systems with invariant lines of total multiplicity eight and with two distinct real infinite singularities . In: Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2015, nr. 1(77), pp. 48-86. ISSN 1024-7696. |
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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica | ||||||
Numărul 1(77) / 2015 / ISSN 1024-7696 /ISSNe 2587-4322 | ||||||
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CZU: 517 | ||||||
Pag. 48-86 | ||||||
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Rezumat | ||||||
In this article we classify a subfamily of differential real cubic systems
possessing eight invariant straight lines, including the line at infinity and including
their multiplicities. This subfamily of systems is characterized by the existence of two
distinct infinite singularities, defined by the linear factors of the polynomial C3(x, y) =
yp3(x, y) − xq3(x, y), where p3 and q3 are the cubic homogeneities of these systems.
Moreover we impose additional conditions related with the existence of triplets and/or
couples of parallel invariant lines. This classification, which is taken modulo the action
of the group of real affine transformations and time rescaling, is given in terms of affine
invariant polynomials. The invariant polynomials allow one to verify for any given real
cubic system whether or not it has invariant straight lines of total multiplicity eight,
and to specify its configuration of straight lines endowed with their corresponding real
singularities of this system. The calculations can be implemented on computer and
the results can therefore be applied for any family of cubic systems in this class, given
in any normal form. |
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Cuvinte-cheie Cubic differential system, configuration of invariant straight lines, multiplicity of an invariant straight line, Group action, affine invari- ant polynomial. |
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