We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. As these series generally have support of order type larger than that of the set of natural numbers, the notion of asymptotic expansion itself needs to be generalized.

中文翻译：

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基于超序列渐近展开的 Ilyashenko 代数

我们构建了一个包含 Ilyashenko 的几乎正则函数无穷远的细菌类以及所有对数-exp-分析细菌的哈代域。此外，这个 Hardy 场中的每个细菌都具有独特的特征，即渐近扩展是 van den Dries 等人定义的 LE 系列。由于这些级数通常具有比自然数集更大的阶类型支持，渐近展开的概念本身需要推广。