# Solution of the 1 : −2 resonant center problem in the quadratic case

Fundamenta Mathematicae (1998)

• Volume: 157, Issue: 2-3, page 191-207
• ISSN: 0016-2736

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## Abstract

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The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $\left(x+{A}_{1}{x}^{2}+{B}_{1}xy+C{y}^{2}\right){\partial }_{x}+\left(-2y+D{x}^{2}+{A}_{2}xy+{B}_{2}{y}^{2}\right){\partial }_{y}$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic properties of polynomial vector fields.

## How to cite

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Fronville, Alexandra, Sadovski, Anton, and Żołądek, Henryk. "Solution of the 1 : −2 resonant center problem in the quadratic case." Fundamenta Mathematicae 157.2-3 (1998): 191-207. <http://eudml.org/doc/212285>.

@article{Fronville1998,
abstract = {The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x + A_1x^2 + B_1xy + Cy^2) ∂_x+(-2y + Dx^2 + A_2xy + B_2y^2)∂_y$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic properties of polynomial vector fields.},
author = {Fronville, Alexandra, Sadovski, Anton, Żołądek, Henryk},
journal = {Fundamenta Mathematicae},
keywords = {resonance; center; quadratic system},
language = {eng},
number = {2-3},
pages = {191-207},
title = {Solution of the 1 : −2 resonant center problem in the quadratic case},
url = {http://eudml.org/doc/212285},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Fronville, Alexandra
AU - Żołądek, Henryk
TI - Solution of the 1 : −2 resonant center problem in the quadratic case
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 191
EP - 207
AB - The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x + A_1x^2 + B_1xy + Cy^2) ∂_x+(-2y + Dx^2 + A_2xy + B_2y^2)∂_y$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic properties of polynomial vector fields.
LA - eng
KW - resonance; center; quadratic system
UR - http://eudml.org/doc/212285
ER -

## References

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1. [1] S. S. Abhyankar and T. T. Moch, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166. Zbl0332.14004
2. [2] J.-C. Faugère, Documentation Hyperdoc de Gb, xmosaic-home http://posso.ibp.fr/Gb.html.
3. [3] J.-P. Françoise and Y. Yomdin, Bernstein inequalities and applications to analytic geometry and differential equations, J. Funct. Anal. 146 (1997) 185-205.
4. [4] A. Fronville, Algorithmic approach to the center problem for 1:-2 resonant singular points of polynomial vector fields, Nonlinearity, submitted.
5. [5] H. Żołądek, The problem of center for resonant singular points of polynomial vector fields, J. Differential Equations 137 (1997), 94-118. Zbl0885.34034

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