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arXiv · 1212.6409

Newton's Method Over Global Height Fields

Xander Faber·Adam Towsley
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Abstract

Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton's method applied to a squarefree polynomial f with K-coefficients will succeed in finding a root of f in the v-adic topology for infinitely many places v of K. Furthermore, we show that if K is a finite extension of the rationals or of the rational function field over a finite field, then the Newton approximation sequence fails to converge v-adically for a positive density of places v.

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Xander Faber, Adam Towsley. 2013-02-13. Newton's Method Over Global Height Fields. https://arxiv.org/abs/1212.6409

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