Algebraic equations with invariant coefficients in qualitative study of the polynomial homogeneous differential systems

Valeriu Baltag

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2003), pp. 13-27.

Résumé

For planar polynomial homogeneous real vector field $X=(P,Q)$ with $\mathrm{deg}(P)=\mathrm{deg}(Q)=n$ some algebraic equations of degree $n+1$ with $GL(2,\mathbb{R})$-invariant coefficients are constructed. A recurrent method for the construction of these coefficients is given. In the generic case each real or imaginary solution $s_i(i=1,2,\dots ,n+1)$ of the main equation is a value of the derivative of the slope function, calculated for the corresponding invariant line. Other constructed equations have, respectively, the solutions $1/s_i$, $1-s_i$, $s_i/(s_i-1)$, $(s_i-1)/s_i$, $1/(1-s_i)$. The equation with the solutions $(n+1)s_i-1$ is called residual equation. If $X$ has real invariant lines, the values and signs of solutions of constructed equations determine the behavior of the orbits in a neighbourhood at infinity. If $X$ has not real invariant lines, it is shown that the necessary and sufficient conditions for the center existence can be expressed through the coefficients of residual equation.

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