The minimal resultant locus

Robert Rumely. "The minimal resultant locus." Acta Arithmetica 169.3 (2015): 251-290. <http://eudml.org/doc/279136>.

@article{RobertRumely2015,
abstract = {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.},
author = {Robert Rumely},
journal = {Acta Arithmetica},
keywords = {minimal resultant locus; potential good reduction},
language = {eng},
number = {3},
pages = {251-290},
title = {The minimal resultant locus},
url = {http://eudml.org/doc/279136},
volume = {169},
year = {2015},
}

TY - JOUR
AU - Robert Rumely
TI - The minimal resultant locus
JO - Acta Arithmetica
PY - 2015
VL - 169
IS - 3
SP - 251
EP - 290
AB - 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.
LA - eng
KW - minimal resultant locus; potential good reduction
UR - http://eudml.org/doc/279136
ER -