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Chebyshev's method for number fields

José Felipe Voloch — 2000

Journal de théorie des nombres de Bordeaux

We give an elementary proof of an explicit estimate for the number of primes splitting completely in an extension of the rationals. The proof uses binomial coefficents and extends Chebyshev's classical approach.

On some subgroups of the multiplicative group of finite rings

José Felipe Voloch — 2004

Journal de Théorie des Nombres de Bordeaux

Let S be a subset of F q , the field of q elements and h F q [ x ] a polynomial of degree d > 1 with no roots in S . Consider the group generated by the image of { x - s s S } in the group of units of the ring F q [ x ] / ( h ) . In this paper we present a number of lower bounds for the size of this group. Our main motivation is an application to the recent polynomial time primality testing algorithm [AKS]. The bounds have also applications to graph theory and to the bounding of the number of rational points on abelian covers of the projective...

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...

Visible Points on Curves over Finite Fields

Igor E. ShparlinskiJosé Felipe Voloch — 2007

Bulletin of the Polish Academy of Sciences. Mathematics

For a prime p and an absolutely irreducible modulo p polynomial f(U,V) ∈ ℤ[U,V] we obtain an asymptotic formula for the number of solutions to the congruence f(x,y) ≡ a (mod p) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.

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