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The minimal resultant locus

Volume 169 / 2015

Robert Rumely Acta Arithmetica 169 (2015), 251-290 MSC: Primary 37P50, 11S82; Secondary 37P05, 11Y40, 11U05. DOI: 10.4064/aa169-3-3

Abstract

Let $K$ be a complete, algebraically closed nonarchimedean valued field, and let $\varphi(z) \in K(z)$ have degree $d \ge 2$. We study how the resultant of $\varphi$ varies under changes of coordinates. For $\gamma \in {\rm GL}_2(K)$, we show that the map $\gamma \mapsto {\rm ord}({\rm Res}(\varphi^\gamma))$ factors through a function ${\rm ordRes}_\varphi(\cdot)$ on the Berkovich projective line, which is piecewise affine and convex up. The minimal resultant is achieved either at a single point in ${\bf P}^1_K$, or on a segment, and the minimal resultant locus is contained in the tree in ${\bf P}^1_K$ spanned by the fixed points and poles of $\varphi$. We give an algorithm to determine whether $\varphi$ has potential good reduction. When $\varphi$ is defined over $\mathbb Q$, the algorithm runs in probabilistic polynomial time. If $\varphi$ has potential good reduction, and is defined over a subfield $H \subset K$, we show there is an extension $L/H$ with $[L:H] \le (d+1)^2$ such that $\varphi$ has good reduction over $L$.

Authors

  • Robert RumelyDepartment of Mathematics
    University of Georgia
    Athens, GA 30602, U.S.A.
    e-mail

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