The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $$f^n\left(z\right)+P_d(z,f)=p_1\left(z\right){\mathrm{e}}^{{\alpha }_1\left(z\right)}+p_2\left(z\right){\mathrm{e}}^{{\alpha }_2\left(z\right)},$$ where $P_d(z,f)$ is a difference-differential polynomial in $f\left(z\right)$ of degree $d\le n-1$ with small functions of $f\left(z\right)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and ${\alpha }_1$, ${\alpha }_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

We investigate the quadratic homogeneous holomorphic vector fields on ${\mathbf{C}}^n$ that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove...

Nous mettons en évidence une obstruction au prolongement d’un germe de champ de vecteurs holomorphe en un champ holomorphe complet. En particulier, on démontre que toute singularité isolée d’un champ holomorphe complet sur une surface complexe possède un deuxième jet non nul.

On démontre qu'une feuille transcendante d'un feuilletage analytique sur une surface fibrée doit intersecter toute courbe algébrique non invariante et non contenue dans une réunion de fibres de la fibration; comme application on montre qu'une équation différentielle algébrique qui possède une solution locale avec une singularité essentielle n'a pas de ramification mobile, ce qui généralise les théorèmes de Malmquist et Yosida.

This paper is devoted to considering the iterated order and the fixed points of some differential polynomials generated by solutions of the differential equation $$f^{{}^{\text{'}\text{'}}}+A_1\left(z\right)f^{{}^{\text{'}}}+A_0\left(z\right)f=F,$$ where $A_1\left(z\right)$, $A_0\left(z\right)$$(\neg \equiv 0)...$

Suppose that $f\left(z\right)$ is a meromorphic or entire function satisfying $P\left(z,f,f^{\prime },\mathrm{\dots },f^{\left(n\right)}\right)=0$ where $P$ is a polynomial in all its arguments. Is there a limitation on the growth of $f$, as measured by its characteristic $T\left(r,f\right)$? In general the answer to this question is not known. Theorems of Gol'dberg, Steinmetz and the author give a positive answer in certain cases. Some illustrative examples are also given.