Valuations on K[x] approaching a fixed irreducible polynomial

2021/06/10 by Novacoski, Josnei, Barnabe, Matheus dos S.
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.05765

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∈ \mathcal V_F and a key polynomial Q∈\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.

Related

• Extensions of a valuation from K to K[x]
• On truncations of valuations
• Valuations with infinite limit-depth
• Valuative trees over valued fields
• Key polynomials and minimal pairs