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Canonical expansions of ℝ𝒬-germs

  • Mostafa Mirabi
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Abstract

Let ℋ𝒬 be the field of unary germs at 0+ definable in the quasianalytic o-minimal structure ℝ𝒬 of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of ℋ𝒬 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 ℝ𝒬 is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic ℝ𝒬-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 ℝ𝒬-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of ℋ𝒬.

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Cite

BibTeXRIS
BibTeX
@misc{mirabi2026canonical,
  title = {{Canonical expansions of $\mathbb R_\mathcal Q$-germs}},
  author = {Mostafa Mirabi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.13189},
  eprint = {2609.13189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13189},
}
RIS
TY  - UNPB
TI  - Canonical expansions of ℝ_𝒬-germs
AU  - Mostafa Mirabi
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.13189
UR  - https://arxiv.org/abs/2609.13189
AB  - Let β„‹_𝒬 be the field of unary germs at 0^+ definable in the quasianalytic o-minimal structure ℝ_𝒬 of Kaiser-Rolin-Speissegger. We construct a canonical ordered differential-field embedding of β„‹_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 ℝ_𝒬 is branching in the sense of Dembner [1]. A nonresonant hyperbolic saddle with divergent Dulac series yields a bounded positive real-analytic ℝ_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 ℝ_𝒬-germ has a canonical formal generalized power-series expansion. Consequently, the convergent germs form a proper ordered differential subfield of β„‹_𝒬.
ER  -