Mathematical Research Letters

Volume 6 (1999)

Number 1

Beyond Liouvillian transcendence

Pages: 31 – 41



C. Camacho

B. Azevedo Scárdua


To a codimension one foliation $\Cal F$ defined by a meromorphic $1$-form $\omega$, one may associate a {\it Godbillon-Vey sequence} $(\omega_j)$,$j=0,1,\dots$, of meromorphic $1$-forms $\omega_j$ with $\omega_0=\omega$. The sequence is said to have {\it finite length} $k$ if $\omega_k\ne 0$ and $\omega_j=0$ for $j >k$. The case $k=0,1$ or $2$ corresponds, respectively, to the case where the foliation $\Cal F$ has additive, affine or projective transverse structure and $k\le 1$ is equivalent to the existence of a Liouvillian first integral. These are the only possible cases where the transverse structures come from an action of a Lie group on $\overline {\Bbb C}$ and a non-trivial model for these foliations is the Riccati differential equation. We propose to go beyond the Lie group transverse structure by studying the case of general $k$ and, for this case, we determine a model differential equation, which generalizes the Riccati equation. We also discuss some other related topics.

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