arXiv subjects
Publications and source records attributed to André Belotto.
Given a real- or complex-analytic singular foliation $θ$ with $n$ first integrals of meromorphic or Darboux type $(f_1,\dots,f_n)$, we prove that there exists a local monomialization of the first integrals. In particular, if $θ$ is generated by the $n$ first integrals, we prove the existence of a local reduction of singularities of $θ$ to monomial singularities.
Let $M$ be an analytic manifold over $\mathbb{R}$ or $\mathbb{C}$, $θ$ a $1$-dimensional Log-Canonical (resp. monomial) singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a resolution of singularities for $\mathcal{I}$ that preserves the Log-Canonicity (resp. monomiality) of the singularities of $θ$. Furthermore, we apply this result to provide a resolution of a family of ideal sheaves when the dimension of the parameter space is equal to the dimension of the ambient space minus one.
We present non-trivial examples of accumulation of secants for orbits (of real analytic three dimensional vector fields) having the origin as only $ω$-limit point. These non-trivial sets have the structure of a proper algebraic variety of $\mathbb{S}^2$ intersected with a cone.
Let $f(x,y) \not\equiv 0$ be a real-analytic planar function. We show that, for almost every $R>0$ there exists an analytic 1-parameter family of vector fields $X_λ$ which has $\{f(x,y)=0\} \cap \bar{B_R((0,0))}$ as a limit periodic set. Furthermore, we show that if $f(x,y)$ is polynomial, then there exists a polynomial family with these properties.