Multivariate Diophantine equations with many solutions

Abstract

Among other things we show that for each n-tuple of positive rational numbers (a 1 , . . . , a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 +• • •+a n x n = 1 with x 1 , . . . , x n S-units are not contained in fewer than exp((4 + o(1))s 1/2 (log s) −1/2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 ≤ m < n, and any sufficiently large s there are integers

Sur l'inégalité de Turán-Kubilius friable

“…Hildebrand a démontré (8) que la fonction r est strictement croissante sur [1; ∞[. Le comportement asymptotique de la fonction r peutêtre déduit de celui de la fonction ξ par le biais de la formule de Alladi-de Bruijn (voir par exemple [27], théorème III.5.13), qui fournit une formule asymptotique pour (v) en fonction de ξ(v).…”

On finding many solutions to S-unit equations by solving linear equations on average

“…For example, in 1984 Evertse [10] used Diophantine approximation methods to give upper bounds, depending only on s (and, in the number field case, also the degree of the field), for the number of S-unit solutions to equations like a + 1 = c. In 1988, Erdős, Stewart and Tijdeman [9] used methods from combinatorial number theory, in particular some results they proved about the largest prime factor of a product of sums, to show the existence of sets S for which a + b = c has many S-unit solutions with a, b, c coprime integers. In 2003, Evertse, Moree, Stewart and Tijdeman [11] gave an extension of Erdős, Stewart and Tijdeman's result to weighted S-unit equations in n variables 1 (as Date: 18th August 2011. The author is supported by a studentship from the Engineering and Physical Sciences Research Council of the United Kingdom.…”

“…The author is supported by a studentship from the Engineering and Physical Sciences Research Council of the United Kingdom. 1 When comparing Evertse, Moree, Stewart and Tijdeman's paper [11] with other results, one should carefully note that they allow the variables in their equations to take rational, and not just integer, values. Their results about the equation a 1 + a 2 + ... + a n = 1, n ≥ 2, with rational variables a i , translate into results about the integer equation a 1 + a 2 + ... + a n = a n+1 .…”

“…Before stating our results, we explain a general strategy for exhibiting many solutions to an S-unit equation. The work of Erdős, Stewart and Tijdeman [9], of Evertse, Moree, Stewart and Tijdeman [11], and of Konyagin and Soundararajan [17] can be seen to fit into this strategy, and our approach is also based upon it.…”

On ideals free of large prime factors

“…The first goal of this paper is to establish a result comparable to (5) in the case when the rational field Q is replaced by an algebraic number field K. Let K be a number field with degree n ≥ 2 and ring of integers O K . For any ideal a of O K , define (6) P (a) = max{N (p) : p|a} where p denotes a prime ideal with norm N (p), and let P (O K ) = 1. Define Ψ K (x, y) by (7) Ψ K (x, y) = |{a : N (a) ≤ x, P (a) ≤ y}| .…”