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Newton’s method over global height fields

Xander FaberAdam Towsley — 2014

Journal de Théorie des Nombres de Bordeaux

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 K x will succeed in finding some 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...

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