标题： 单值奇点的主Bautin理想与反积分因子 显示英文标题

标题： Principal Bautin ideal of monodromic singularities with inverse integrating factors

摘要： 我们分析与平面向量场的解析族中单值奇点相关的庞加莱映射$\Pi$的结构。 我们在两个假设下进行工作。 第一个假设是，该族在通过该族的牛顿图固定的一个加权极坐标爆破后，可以展开为以奇点为中心的洛朗级数的逆积分因子。 第二个假设是我们将分析限制在单值参数空间的一个子集，以保证不存在角速度为零的局部曲线。 结论是，$\Pi$的渐近 Dulac 展开不包含对数项，实际上它接受一个形式幂级数展开，其中包含一个独立的广义庞加莱-李雅普诺夫量，该量在一些显式条件下可以计算。 此外，我们还给出了保证$\Pi$解析性的条件，在这种情况下，我们证明了 Bautin 理想是主理想，因此该奇点在族内的扰动下的环性为零。 显示英文摘要

摘要： We analyze the structure of the Poincar\'e map $\Pi$ associated to a monodromic singularity of an analytic family of planar vector fields. We work under two assumptions. The first one is that the family possesses an inverse integrating factor that can be expanded in Laurent series centered at the singularity after a weighted polar blow-up fixed by the Newton diagram of the family. The second one is that we restrict our analysis to a subset of the monodromic parameter space that assures the non-existence of local curves with zero angular speed. The conclusions are that the asymptotic Dulac expansion of $\Pi$ does not contain logarithmic terms, indeed it admits a formal power series expansion with a unique independent generalized Poincar\'e-Lyapunov quantity, which can be computed under some explicit conditions. Moreover we also give conditions that guarantee the analyticity of $\Pi$, in which case we show that the Bautin ideal is principal and therefore the cyclicity of the singularity with respect to perturbation within the family is zero.