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

• Autorzy: Alexandra Fronville , Anton P. Sadovski , Henryk Żołądek
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1998
• Tom: 157
• Numer: 2-3
• Strony: 191-207

Daty

wydano

1998

otrzymano

1997-10-25

Twórcy

autor

Alexandra Fronville

• UMR 9994 du CNRS, Université P. et M. Curie 4, pl. Jussieu, 75252 Paris Cedex 05, France

autor

Anton P. Sadovski

• Department of Mathematics, Grodno State University, Orzeszko St. 22, 230023 Grodno, Belarus

autor

Henryk Żołądek

• Institute of Mathematics, University of Warsaw, Banacha 2 02-097 Warszawa, Poland

Bibliografia

• [1] S. S. Abhyankar and T. T. Moch, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148-166.
• [2] J.-C. Faugère, Documentation Hyperdoc de Gb, xmosaic-home http://posso.ibp.fr/Gb.html.
• [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] A. Fronville, Algorithmic approach to the center problem for 1:-2 resonant singular points of polynomial vector fields, Nonlinearity, submitted.
• [5] H. Żołądek, The problem of center for resonant singular points of polynomial vector fields, J. Differential Equations 137 (1997), 94-118.