Canonical expansions of ${\mathbb{R}}_{\mathcal{Q}}$-germs

Abstract

Let ${\mathcal{H}}_{\mathcal{Q}}$ be the field of unary germs at $0^{+}$ definable in the quasianalytic o-minimal structure ${\mathbb{R}}_{\mathcal{Q}}$ of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of ${\mathcal{H}}_{\mathcal{Q}}$ into the field of natural-support generalized Laurent series. Every unary germ has a unique such asymptotic expansion, and bounded germs have expansions with only nonnegative exponents. A germ admits a convergent generalized Laurent-series representation exactly when its canonical expansion converges; in that case, every convergent representation with well-ordered support is canonical. We also prove that ${\mathbb{R}}_{\mathcal{Q}}$ is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic ${\mathbb{R}}_{\mathcal{Q}}$-definable germ whose canonical generalized power series diverges. This gives a negative answer to a question of Dembner asking whether every unary germ definable in a branching structure admits a convergent generalized power-series representation. Nevertheless, every bounded unary ${\mathbb{R}}_{\mathcal{Q}}$-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of ${\mathcal{H}}_{\mathcal{Q}}$.

arXiv comment15 pages