Geodesic
On the upper bound of the number of functionally independent focal quantities of the Lyapunov differential system
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2019), pp. 99-112 
Mihail Popa; Victor Pricop. On the upper bound of the number of functionally independent focal quantities of the Lyapunov differential system. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2019), pp. 99-112. http://geodesic.mathdoc.fr/item/BASM_2019_2_a5/
@article{BASM_2019_2_a5,
     author = {Mihail Popa and Victor Pricop},
     title = {On the upper bound of the number of functionally independent focal quantities of the {Lyapunov} differential system},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {99--112},
     year = {2019},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2019_2_a5/}
}
TY  - JOUR
AU  - Mihail Popa
AU  - Victor Pricop
TI  - On the upper bound of the number of functionally independent focal quantities of the Lyapunov differential system
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2019
SP  - 99
EP  - 112
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/BASM_2019_2_a5/
LA  - en
ID  - BASM_2019_2_a5
ER  - 
%0 Journal Article
%A Mihail Popa
%A Victor Pricop
%T On the upper bound of the number of functionally independent focal quantities of the Lyapunov differential system
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2019
%P 99-112
%N 2
%U http://geodesic.mathdoc.fr/item/BASM_2019_2_a5/
%G en
%F BASM_2019_2_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

Denote by $N_1=2\sum\limits_{i=1}^{\ell}(m_i+1)+2$ the maximal possible number of non-zero coefficients of the Lyapunov differential system $\dot{x}= y+\sum\limits_{i=1}^{\ell}P_{m_i}(x,y)$, $\dot{y}= -x+\sum\limits_{i=1}^{\ell}Q_{m_i}(x,y)$, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i$ with respect to $x$ and $y$, and $1$ $(\ell\infty)$. Then the upper bound of functionally independent focal quantities in the center and focus problem of considered system does not exceed $N_1-1$.

[1] Popa M. N., Pricop V. V., “Applications of algebraic methods in solving the center-focus problem”, Bul. Acad. Ştiinţe Repub. Moldova, Matematica, 2013, no. 1(71), 45–71, arXiv: 1310.4343 [math.DS] | MR | Zbl

[2] Popa M. N., Pricop V. V., The center and focus problem: algebraic solutions and hypotheses, Academy of Sciences of Moldova, Chişinău, 2018 (in Russian) | MR

[3] Popa M. N., Algebraic methods for differential systems, The Flower Power Edit, 2004 (in Romanian) | Zbl

[4] Sibirsky K. S., Introduction to the algebraic theory of invariants of differential equations, Translated from Russian, Nonlinear Science: Theory and Applications, Manchester University Press, Manchester, 1988 | MR | Zbl

[5] Pontryagin L. S., Ordinary differential equations, Addison Wesley Publishing Company, London, 1962, vi+298 pp. | MR | Zbl

[6] Sibirsky K. S., Algebraic Invariants of Differential Equations and Matrices, Shtiintsa, Kishinev, 1976 (in Russian) | MR

[7] Sadovsky A. P., Polynomial ideals and varieties, BSU, Minsk, 2008 (in Russian)