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Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, 2013, Number 1, Pages 45–71
(Mi basm328)
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This article is cited in 4 scientific papers (total in 4 papers)
Research articles
Applications of algebraic methods in solving the center-focus problem
M. N. Popa, V. V. Pricop Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, 5, Academiei str., Chişinău, MD-2028 Moldova
Abstract:
The nonlinear differential system $\dot x=\sum_{i=0}^\ell P_{m_i}(x,y)$, $\dot y=\sum_{i=0}^\ell Q_{m_i}(x,y)$ is considered, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i\geq1$ in $x$ and $y$, $m_0=1$. The set $\{1,m_i\}_{i=1}^\ell$ consists of a finite number $(l<\infty)$ of distinct integer numbers. It is shown that the maximal number of algebraically independent focal quantities that take part in solving the center-focus problem for the given differential system with $m_0=1$, having at the origin of coordinates a singular point of the second type (center or focus), does not exceed $\varrho=2(\sum_{i=1}^\ell m_i+\ell)+3$. We make an assumption that the number $\omega$ of essential conditions for center which solve the center-focus problem for this differential system does not exceed $\varrho$, i.e. $\omega\leq\varrho$.
Keywords and phrases:
differential systems, the center-focus problem, focal quantities, Sibirsky graded algebras, Hilbert series, Krull dimension, Lie algebras of operators.
Received: 10.11.2012
Citation:
M. N. Popa, V. V. Pricop, “Applications of algebraic methods in solving the center-focus problem”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2013, no. 1, 45–71
Linking options:
https://www.mathnet.ru/eng/basm328 https://www.mathnet.ru/eng/basm/y2013/i1/p45
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