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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...

On the number of places of convergence for Newton’s method over number fields

Xander FaberJosé Felipe Voloch — 2011

Journal de Théorie des Nombres de Bordeaux

Let f be a polynomial of degree at least 2 with coefficients in a number field K , let x 0 be a sufficiently general element of K , and let α be a root of f . We give precise conditions under which Newton iteration, started at the point x 0 , converges v -adically to the root α for infinitely many places v of K . As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v -adically to any given root of f for infinitely many places v . We also conjecture that...

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