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Some remarks about \( \rho \)-regularity for real analytic maps

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In this paper, we discuss the concept of \(\rho \)-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor tube fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function \(\rho \), which guarantees \(\rho \)-regularity. Consequently, both conditions are crucial for the presence of fibration structures. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.

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  1. The latter condition is known as the Milnor condition (b) at the origin.

  2. Despite being classical results, proofs are generally not found in the literature, except perhaps in more recent works, such as [58, Theorem 4.14, Proposition 1.2.26, Proposition 4.3.2] and [43, Theorem 2.2, Lemma 2.3, Proposition 3.1, and Proposition 4.2], where the authors define and consider \(\rho \)-regularity, the Milnor condition (b), and the existence of Thom stratification in a more general context where the dimension of the discriminant is positive.

  3. See Theorem 2.4.

  4. Or simply, G is Thom regular.

  5. In other words, if \(W_\alpha \) and \(W_\beta \) are strata of \(\mathbb {W}\) such that \(W_\alpha \subset V_G\) and \(W_\alpha \subset {\overline{W}}_\beta \), then the pair \((W_\beta , W_\alpha )\) satisfies the Thom \((\mathrm{{a}}_G)\)-regularity condition.

  6. See [29, Theorem 1.2.1]

  7. See Section 4.1 for definition.

  8. See Remark 4.5 for definition.

  9. In fact, it was shown in [52, Theorem 2.3] that the hypothesis “\(\mathrm{{Disc\hspace{2pt}}}(f,g) \) only contains curves tangent to the coordinate axes” is equivalent to \( \mathrm{{Disc\hspace{2pt}}}f{\bar{g}} = \{0\} \)

  10. Now, since \(\langle (0,0,\pm 1), (0, \pm 1, 0) \rangle = 0\), we have \((0,0,\pm 1) \in {{\mathcal {T}}}^{\perp }\). In other words, \({{\mathcal {T}}}^{\perp } \not \subset \left( {{\mathcal {T}}}_{p}W_i\right) ^{\perp }\). Consequently, \(\text {T}_{p}W_i \not \subset {{\mathcal {T}}}\), as previously shown in Example 3.12.

  11. Actually, as “any Whitney stratification” we mean “any reduced Whitney stratification” in the following sense. Reduced means: let \( \mathbb {W}=\{W_\alpha \} \) be a Whitney stratification and suppose \( W_1 \) and \( W_2 \) are two strata such that \( W_2 \subset \overline{W_1} \) and \( W_1 \cup W_2 \) is also a smooth manifold so that two strata can be put one keeping regularity of Whitney stratification.

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Acknowledgements

Maico Ribeiro is Funded by CNPQ/MCTI grant number 408147/2023-7 and CAPES grant number 88887.909401/2023-00. Ivan Santamaria is Funded by CAPES 88887.500400/2020-00. Thiago da Silva is Funded by CAPES grant number 88887.909401/2023-00 and CAPES grant number 88887.897201/2023-00.

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    Maico Ribeiro, Ivan Santamaria & Thiago da Silva

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Ribeiro, M., Santamaria, I. & da Silva, T. Some remarks about \( \rho \)-regularity for real analytic maps. Res Math Sci 11, 40 (2024). https://doi.org/10.1007/s40687-024-00453-y

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