# The minimal resultant locus

Acta Arithmetica (2015)

- Volume: 169, Issue: 3, page 251-290
- ISSN: 0065-1036

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topRobert 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 -

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