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Minimal polynomial basis of centroaffine comitants of the two-dimensional system of differential equations with nonlinearities of the fourth degree

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Τίτλος: Minimal polynomial basis of centroaffine comitants of the two-dimensional system of differential equations with nonlinearities of the fourth degree
Συγγραφείς: Ciubotaru, S.S., Calin, I.T., Kalin, Y. .
Πηγή: International Conference of Young Researchers (Ediția 10)
Έτος έκδοσης: 2012
Θεματικοί όροι: system of differential equations, minimal polynomial basis, transvectant, comitants, invariants
Περιγραφή: Let us consider the system of differential equations with nonlinearities of the fourth degree݀nbsp; ݔ௝݀ ݐ =ܲ ଵ ௝ ݔଵ, ݔଶ +ܲ ସ ௝ ݔଵ, ݔଶ, ݆ = 1, 2, 1 whereܲnbsp; ଵ ௝ andܲnbsp; ସ ௝ are homogeneous polynomials of degree 1 and 4, respectively, in ݔଵ and ݔଶ with real coefficients. We say that a homogeneous ܮܩ2, ℝ-comitant centroaffine comitant Ka; x has the character r; g; d if it has the weight g, the degree d with respect to the coefficients of system 1 and the degree r with respect to the coordinates of the vector ݔ isin; ℝଶ [1]. Definition 1. The set S of comitants respectively, invariants is called a polynomial basis of comitants respectively, invariants for system 1 with respect to a group Q if any comitant respectively, invariant of system 1 with respect to the group Q can be expressed in the form of a polynomial of elements of the set S. Definition 2. A polynomial basis of comitants respectively, invariants for system 1 with respect to a group Q is called minimal if by the removal from it of any comitant respectively, invariant it ceases to be a polynomial basis. The theory of algebraic invariants and comitants for polynomial autonomous systems of differential equations has been developed by C. Sibirschi [1, 2] and his disciples. One of the important problems concerning this theory is the construction of minimal polynomial bases of the invariants and comitants of the mentioned systems, with respect to different subgroups of the affine group of the transformations of their phase planes, in particular with respect to the subgroup GL2;R. Some important results in this direction are obtained by academician C. Sibirschi [1, 2] and N.Vulpe [3]. We remark, that polynomial bases for different particular combinations of homogeneous polynomialsܲnbsp; ௠ ௝ ݔଵ, ݔଶ j = 1, 2; m = 0, 1, 2, 3 in system 1 are also considered by E. Gasinskaya-Kirnitskaya, Dang Dinh Bich, D. Boularas, M. Popa, V. Ciobanu, V. Danilyuk, E. Naidenova. We shall consider the following polynomials:ܴ ௜ =ܲ ௜ଵݔଶ minus;ܲ ௜ଶݔଵ ,ܵ ௜ = 1݅ ቆ߲ܲ ௜ଵ߲ ݔଵ +߲ܲ ௜ଶ߲ ݔଶ ...
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Relation: https://ibn.idsi.md/vizualizare_articol/231919
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  Data: Minimal polynomial basis of centroaffine comitants of the two-dimensional system of differential equations with nonlinearities of the fourth degree
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  Data: International Conference of Young Researchers (Ediția 10)
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  Data: Let us consider the system of differential equations with nonlinearities of the fourth degree݀nbsp; ݔ௝݀ ݐ =ܲ ଵ ௝ ݔଵ, ݔଶ +ܲ ସ ௝ ݔଵ, ݔଶ, ݆ = 1, 2, 1 whereܲnbsp; ଵ ௝ andܲnbsp; ସ ௝ are homogeneous polynomials of degree 1 and 4, respectively, in ݔଵ and ݔଶ with real coefficients. We say that a homogeneous ܮܩ2, ℝ-comitant centroaffine comitant Ka; x has the character r; g; d if it has the weight g, the degree d with respect to the coefficients of system 1 and the degree r with respect to the coordinates of the vector ݔ isin; ℝଶ [1]. Definition 1. The set S of comitants respectively, invariants is called a polynomial basis of comitants respectively, invariants for system 1 with respect to a group Q if any comitant respectively, invariant of system 1 with respect to the group Q can be expressed in the form of a polynomial of elements of the set S. Definition 2. A polynomial basis of comitants respectively, invariants for system 1 with respect to a group Q is called minimal if by the removal from it of any comitant respectively, invariant it ceases to be a polynomial basis. The theory of algebraic invariants and comitants for polynomial autonomous systems of differential equations has been developed by C. Sibirschi [1, 2] and his disciples. One of the important problems concerning this theory is the construction of minimal polynomial bases of the invariants and comitants of the mentioned systems, with respect to different subgroups of the affine group of the transformations of their phase planes, in particular with respect to the subgroup GL2;R. Some important results in this direction are obtained by academician C. Sibirschi [1, 2] and N.Vulpe [3]. We remark, that polynomial bases for different particular combinations of homogeneous polynomialsܲnbsp; ௠ ௝ ݔଵ, ݔଶ j = 1, 2; m = 0, 1, 2, 3 in system 1 are also considered by E. Gasinskaya-Kirnitskaya, Dang Dinh Bich, D. Boularas, M. Popa, V. Ciobanu, V. Danilyuk, E. Naidenova. We shall consider the following polynomials:ܴ ௜ =ܲ ௜ଵݔଶ minus;ܲ ௜ଶݔଵ ,ܵ ௜ = 1݅ ቆ߲ܲ ௜ଵ߲ ݔଵ +߲ܲ ௜ଶ߲ ݔଶ ...
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      – Text: English
    Subjects:
      – SubjectFull: system of differential equations
        Type: general
      – SubjectFull: minimal polynomial basis
        Type: general
      – SubjectFull: transvectant
        Type: general
      – SubjectFull: comitants
        Type: general
      – SubjectFull: invariants
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      – TitleFull: Minimal polynomial basis of centroaffine comitants of the two-dimensional system of differential equations with nonlinearities of the fourth degree
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