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1

The minimal resultant locus

100%

Rumely R.

Acta Arithmetica

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2015

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tom 169

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nr 3

251-290

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.

2

The Fekete-Szegő theorem with splitting conditions: Part II

100%

Rumely R.

Acta Arithmetica

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2002

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tom 103

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nr 4

347-410

3

The Fekete-Szegő theorem with splitting conditions: Part I

100%

Rumely R.

Acta Arithmetica

|

2000

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tom 93

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nr 2

99-116

Ograniczanie wyników

3 Acta Arithmetica

3 Rumely R.

1 2015

1 2002

1 2000

The minimal resultant locus

The Fekete-Szegő theorem with splitting conditions: Part II

The Fekete-Szegő theorem with splitting conditions: Part I