Newton's Method Over Global Height Fields
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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2012
- DOI:
- 10.48550/arXiv.1212.6409
- arXiv:
- arXiv:1212.6409
- Bibcode:
- 2012arXiv1212.6409F
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Dynamical Systems