Principal Poincaré-Pontryagin function associated to polynomial perturbations of a product of \((d+1)\) straight lines
The paper studies the displacement map (associated to a continuous family of ovals) \[ D(t,\varepsilon)=\varepsilon M_1(t)+\varepsilon^2 M_2(t) +\varepsilon^3 M_3(t)+ \ldots \] related to small polynomial perturbations \(dF+\varepsilon \omega=0\) in the case when the Hamiltonian function \(F(x,y)\) is a product of \(d+1\) linear functions \(f_i(x,y)=a_ix+b_iy+c_i\), \(i=0,1,\ldots,d\) and assuming that the corresponding straight lines \(\ell_i: f_i(x,y)=0\) are in a generic position. Then the Hamiltonian vector field has \(a_1=\frac12d(d-1)\) centers and \(a_2=\frac12d(d+1)\) saddles. The main result states that if \(k\geq 2\) and \(M_1=\ldots=M_{k-1}\equiv 0\), \(M_k(t)\not\equiv 0\), then \(M_k(t)\) belongs to a \({\mathbb C}[t,1/t]\)-module generated by \(2a_1\) Abelian integrals \(I_i(t)\) and \(a_1\) non-Abelian integrals \(I_{i,j}^*(t)=\int_{\delta(t)}\varphi_id\varphi_j\), \(1\leq i<j\leq d\) containing logarithmic terms.
scientific article; zbMATH DE number 5529531