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Canonical expansions of $\mathbb R_{\mathcal Q}$-germs — arXiv:2609.13189

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.13189
License (as archived): CC0-1.0
Archived: 2026-10-16T01:38:19.844Z
SHA-256 of the archived text: 8a93808b669999cb1eaf35705c36ef3a8e17732aa746d9568a0c303103a3db36
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Canonical expansions of $\mathbb R_{\mathcal Q}$-germs — arXiv:2609.13189 — arXiv (en), https://arxiv.org/abs/2609.13189 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ca3deaea8a7a (Metraipolis archive dc_ca3deaea8a7a, sha256 8a93808b669999cb1eaf35705c36ef3a8e17732aa746d9568a0c303103a3db36)

BibTeX

@misc{dcca3deaea8a7a,
  title = {Canonical expansions of $mathbb R_mathcal Q$-germs — arXiv:2609.13189},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.13189},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ca3deaea8a7a; sha256 8a93808b669999cb1eaf35705c36ef3a8e17732aa746d9568a0c303103a3db36},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.13189. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $\mathcal{H}_{\mathcal{Q}}$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-15 in math.LO).
Categorie arXiv: Logic (math.LO) ; Dynamical Systems (math.DS).
Pagina del lavoro: https://arxiv.org/abs/2609.13189 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T01:22:41.469Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-15?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).