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The minimal resultant locus

Robert Rumely — 2015

Acta Arithmetica

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 o r d ( R e s ( φ γ ) ) factors through a function o r d R e s φ ( · ) on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in P ¹ K , or on a segment, and the minimal resultant locus is contained in the tree in P ¹ K spanned by the fixed points and poles...

Equidistribution of Small Points, Rational Dynamics, and Potential Theory

Matthew H. BakerRobert Rumely — 2006

Annales de l’institut Fourier

Given a rational function ϕ ( T ) on 1 of degree at least 2 with coefficients in a number field k , we show that for each place v of k , there is a unique probability measure μ ϕ , v on the Berkovich space Berk , v 1 / v such that if { z n } is a sequence of points in 1 ( k ¯ ) whose ϕ -canonical heights tend to zero, then the z n ’s and their Gal ( k ¯ / k ) -conjugates are equidistributed with respect to μ ϕ , v . The proof uses a polynomial lift F ( x , y ) = ( F 1 ( x , y ) , F 2 ( x , y ) ) of ϕ to construct a two-variable Arakelov-Green’s function g ϕ , v ( x , y ) for each v . The measure μ ϕ , v is obtained by...

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