Displaying similar documents to “Lehmer’s conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves”

Power-free values, large deviations, and integer points on irrational curves

Harald A. Helfgott (2007)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let f ∈ ℤ [ x ] be a polynomial of degree d ≥ 3 without roots of multiplicity d or ( d - 1 ) . Erdős conjectured that, if f satisfies the necessary local conditions, then f ( p ) is free of ( d - 1 ) th powers for infinitely many primes p . This is proved here for all f with sufficiently high entropy. The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations. ...

A constant in pluripotential theory

Zbigniew Błocki (1992)

Annales Polonici Mathematici

Similarity:

We compute the constant sup ( 1 / d e g P ) ( m a x S l o g | P | - ∫ S l o g | P | d σ ) : P a polynomial in ℂ n , where S denotes the euclidean unit sphere in ℂ n and σ its unitary surface measure.

Approximation of values of hypergeometric functions by restricted rationals

Carsten Elsner, Takao Komatsu, Iekata Shiokawa (2007)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We compute upper and lower bounds for the approximation of hyperbolic functions at points 1 / s ( s = 1 , 2 , ⋯ ) by rationals x / y , such that x , y satisfy a quadratic equation. For instance, all positive integers x , y with y ≡ 0 ( mod 2 ) solving the Pythagorean equation x 2 + y 2 = z 2 satisfy | y sinh ( 1 / s ) - x | ≫ log log y log y . Conversely, for every s = 1 , 2 , ⋯ there are infinitely many coprime integers x , y , such that | y sinh ( 1 / s ) - x | ≪ log log y log y and x 2 + y 2 = z 2 hold simultaneously for some integer z . A generalization to the approximation of h ( e 1 / s ) for rational...