The minimal resultant locus

Robert Rumely¹

¹ Department of Mathematics University of Georgia Athens, GA 30602, U.S.A.

Acta Arithmetica, Tome 169 (2015) no. 3, pp. 251-290

Résumé

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 $L$.

Détail

DOI : 10.4064/aa169-3-3

Keywords: complete algebraically closed nonarchimedean valued field varphi have degree study resultant varphi varies under changes coordinates gamma map gamma mapsto ord res varphi gamma factors through function ordres varphi cdot berkovich projective line which piecewise affine convex minimal resultant achieved either single point segment minimal resultant locus contained tree spanned fixed points poles varphi algorithm determine whether varphi has potential reduction varphi defined mathbb algorithm runs probabilistic polynomial time varphi has potential reduction defined subfield subset there extension varphi has reduction