Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group
The paper contains a detailed discussion of the general fact that if the functions \(y_1(t)\), \(y_2(t)\), \(y_n(t)\), real analytic on \([a,b]\), constitute a fundamental system of solutions for a linear differential equation with real meromorphic coefficients and an irreducible monodromy group, then the number of isolated zeroes on \([a,b]\), of any combination \(f(t)=\sum^n_{j,r=1} P_{jr}(t)Y^{(r-1)}_j(t)\) with real polynomial coefficients \(P_{jr}(t)\) of degree \(\leq d\), can be at most \(2^{2^{O(d)}}\) with constant \(O(d)\) depending on the linear equation and the segment \([a,b]\) only. Then, it is proved that for a generic real polynomial \(H(x,y)\) in two variables and a polynomial 1-form \(\omega=P(x,y)dx+Q(x,y)dy\) of degree \(d=\max(\deg P,\deg Q)\) the number of real zeroes of the complete Abelian integral \(I(t)=\phi\omega\) over ovals of the curve \(\{H=t\}\) on any compact segment \(t\in(a,b)\subset\mathbb{R}\) not containing critical points of the Hamiltonian \(H\) does not exceed \(2^{2^{O(d)}}\) as \(d\to\infty\) uniformly over all forms.
scientific article; zbMATH DE number 871786