A problem of relative cohomology for algebraic forms of \(\mathbb{R}^ 2\) (Q1913478)

The second part of the Hilbert 16th problem is a very well known and difficult question, extensively developed during the last 10 years, especially in France and in Eastern Europe. The present work is outstanding in what concerns quadratic centres, especially in its approach, initiated in Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, No. 3, 403-424 (1993; Zbl 0805.32007) by the author and \textit{D. Cerveau} (Rennes). The Hilbert 16th problem is all about the non-accumulation of limit cycles for planar, analytic, analytically parametrized families of vector fields. Introductory papers are numerous (the reviewer, Françoise, Roussarie ...), there is also an excellent new book by J. P. Françoise (1995). The problem is still unsolved, even in the case of quadratic polynomials. This is the case dealt with by the author. In \(\mathbb{R}^2\) or \(\mathbb{C}^2\) we can obviously replace analytic vector fields by analytic 1-forms. This latter approach is chosen. The author concentrates his attention on the deformations of a ``centre'' (singular point having only periodic orbits near it). The centres of quadratic forms were classified by Dulac in 1908. The author chooses the class, for which the prime integral is \((x + \alpha)^\alpha (y + \beta)^\beta (y + x - 1)\). The quadratic form \(\omega_0\) with such a centre is then subject to analytic deformations \(\omega_\varepsilon\) and their behaviour is studied. Obviously, each form \(\omega_0\) of this type defines an algebraic foliation \({\mathcal F}\) of \(\mathbb{R}^2 \backslash \text{sing} \omega\). We say that a 1-form \(\omega\) is of zero periods on \({\mathcal F}\) if its integral vanishes on every cycle of \({\mathcal F}\). We say that a 1-form \(\omega\) is exact \(\bmod {\mathcal F}\) if it is exact on every leaf of \({\mathcal F}\), the primitives being independent of the leaf. It is easy to see (via the work of \textit{K. Saito} [Ann. Inst. Fourier 26, No. 2, 165-170 (1976; Zbl 0338.13009)]) that a 1-form which is exact \(\bmod {\mathcal F}\) is of zero periods. But is every 1-form verifying our conditions and of zero periods exact \(\bmod {\mathcal F}\)? The work of the author and \textit{D. Cerveau} [loc. cit.] shows that the answer is yes under some specific conditions (in \(\mathbb{C}^2)\) and the present work generalizes it to the case of quadratic centres in \(\mathbb{R}^2\), with some additional conditions, and then to algebraic foliations ``of logarithmic nature'' (theorems 1.1 and 2.1). Despite the title, somewhat frightening for the non-experienced, the work is easy to follow, because well explained on every level, however high. The work is a very interesting contribution to the Hilbert 16th problem (second part) and is very pleasant to read, despite its depth and level.