Abstract
We generalize notions and results obtained by Amice for regular compact subsets S of a local field K and extended by Bhargava to general compact subsets of K. Considering any ultrametric valued field K and subsets S that are regular in a generalized sense (but not necessarily compact), we show that they still have strong properties such as having v-orderings \({\{a_n\}_{n\geq0}}\) which satisfy a generalized Legendre formula, which are very well ordered and well distributed sequences in the sense of Helsmoortel and which remain v-orderings when a finite number of the initial terms of the sequence are deleted.
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Chabert, JL., Evrard, S. & Fares, Y. Regular subsets of valued fields and Bhargava’s v-orderings. Math. Z. 274, 263–290 (2013). https://doi.org/10.1007/s00209-012-1069-x
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DOI: https://doi.org/10.1007/s00209-012-1069-x
Keywords
- Integer-valued polynomials
- Valued fields
- Regular subsets
- Generalized Legendre formula
- Strong v-orderings
- Integer-valued divided differences