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arXiv · 2106.05765

Valuations on $K[x]$ approaching a fixed irreducible polynomial

Josnei Novacoski·Matheus dos S. Barnabe
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Abstract

For a fixed irreducible polynomial $F$ we study the set $\mathcal V_F$ of all valuations on $K[x]$ bounded by valuations whose support is $(F)$. The first main result presents a characterization for valuations in $\mathcal V_F$ in terms of their graded rings. We also present a result which gives, for a fixed $ν\in \mathcal V_F$ and a key polynomial $Q\in{\rm KP}(ν)$, the maximum value that augmented valuations in $\mathcal V_F$ can assume on $Q$. This value is presented explicitly in terms of the slopes of the Newton polygon of $F$ with respect to $Q$. Finally, we present some results about Artin-Schreier extensions that illustrate the applications that we have in mind for the results in this paper.

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Josnei Novacoski, Matheus dos S. Barnabe. 2021-10-26. Valuations on $K[x]$ approaching a fixed irreducible polynomial. https://arxiv.org/abs/2106.05765

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