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On Regular Mac Lane-Vaquié chains

  • Nikita Dwivedi
  • Anuj Bishnoi
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Abstract

Let (Kh,vh) be a Henselization of a valued field (K,v), μ any extension of v to K[x], and μh the canonical extension of μ and vh to Kh[x]. In this paper, we study two new invariants associated with a Mac Lane-Vaquié chain (MLV) of μ, namely, the defect of each augmentation step and the regularity of an MLV chain of μ. This, in turn, extends the notion of regularity of a complete sequence of abstract key polynomials for μ to the regularity of μ itself. We also prove that the regularity of μ is a sufficient condition for the equality of the depths of μ and μh. Furthermore, it is a necessary and sufficient condition for the equality of the defect and relative gap of each augmentation step of an MLV chain of μ. Finally, we show that if μ has finite degree, then the regularity of μ is equivalent to $ \operatorname{deg}(μ)=\operatorname{deg}(μ^h)$.

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Cite

BibTeXRIS
BibTeX
@misc{dwivedi2026regular,
  title = {{On Regular Mac Lane-Vaquié chains}},
  author = {Nikita Dwivedi and Anuj Bishnoi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23443},
  eprint = {2609.23443},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23443},
}
RIS
TY  - UNPB
TI  - On Regular Mac Lane-Vaquié chains
AU  - Nikita Dwivedi
AU  - Anuj Bishnoi
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23443
UR  - https://arxiv.org/abs/2609.23443
AB  - Let (K^h,v^h) be a Henselization of a valued field (K,v), μ any extension of v to K[x], and μ^h the canonical extension of μ and v^h to K^h[x]. In this paper, we study two new invariants associated with a Mac Lane-Vaquié chain (MLV) of μ, namely, the defect of each augmentation step and the regularity of an MLV chain of μ. This, in turn, extends the notion of regularity of a complete sequence of abstract key polynomials for μ to the regularity of μ itself. We also prove that the regularity of μ is a sufficient condition for the equality of the depths of μ and μ^h. Furthermore, it is a necessary and sufficient condition for the equality of the defect and relative gap of each augmentation step of an MLV chain of μ. Finally, we show that if μ has finite degree, then the regularity of μ is equivalent to $ \operatorname{deg}(μ)=\operatorname{deg}(μ^h)$.
ER  -