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

Volume 157 / 1998

Alexandra Fronville, Anton P. Sadovski, Henryk Żołądek Fundamenta Mathematicae 157 (1998), 191-207 DOI: 10.4064/fm-157-2-3-191-207

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.

Authors

  • Alexandra Fronville
  • Anton P. Sadovski
  • Henryk Żołądek

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