arXiv subjects
Publications and source records attributed to Edileno de Almeida Santos.
We prove a Darboux-Jouanolou type theorem on the algebraic integrability of polynomial differential $r$-forms over arbitrary fields ($r\geq 1$). We also investigate the Darboux's method for producing integrating factors.
Let $K$ be a field of characteristic $ p>0$ and $ω$ be an $r$-form in $ K^n$. In this case, differently of fields of characteristic zero, the Poincaré Lemma is not true because there are closed $ r$-forms that are not exact. We present here a definition of a $p$-closed $r$-forms and a version of the Poincaré Lemma that is valid for $p$-closed polynomial or rational $r$-forms on $ K^n$ and, as a consequence, the de Rham cohomology modules of $ K^n$ are not trivial.
We prove a Darboux-Jouanolou type theorem on the algebraic integrability of differential 1-forms over arbitrary fields.
The Separatrix Theorem of C. Camacho and P. Sad guarantees the existence of invariant curve (separatrix) passing through the singularity of germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.
We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.