The minimal resultant locus

• Autorzy: Robert Rumely
• Języki publikacji: EN

Abstrakty

EN

Let K be a complete, algebraically closed nonarchimedean valued field, and let φ(z) ∈ K(z) have degree d ≥ 2. We study how the resultant of φ varies under changes of coordinates. For γ ∈ GL₂(K), we show that the map $γ ↦ ord(Res(φ^γ))$ factors through a function $ordRes_φ(·)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in $P¹_K$, or on a segment, and the minimal resultant locus is contained in the tree in $P¹_K$ spanned by the fixed points and poles of φ. We give an algorithm to determine whether φ has potential good reduction. When φ is defined over ℚ, the algorithm runs in probabilistic polynomial time. If φ has potential good reduction, and is defined over a subfield H ⊂ K, we show there is an extension L/H with [L:H] ≤ (d+1)² such that φ has good reduction over L.

Kategorie tematyczne

• 11U05: Decidability
• 11S82: Non-Archimedean dynamical systems \SeeMainly{}
• 37P50: Dynamical systems on Berkovich spaces
• 37P05: Polynomial and rational maps
• 11Y40: Algebraic number theory computations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 169
• Numer: 3
• Strony: 251-290

Daty

wydano

2015

Twórcy

autor

Robert Rumely

• Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.

Identyfikatory

DOI

10.4064/aa169-3-3