On the Bateman-horn Conjecture about Polynomial Rings
نویسندگان
چکیده
Given a power q of a prime number p and “nice” polynomials f1, . . . , fr ∈ Fq[T, X] with r = 1 if p = 2, we establish an asymptotic formula for the number of pairs (a1, a2) ∈ Fq such that f1(T, a1T + a2), . . . , fr(T, a1T + a2) are irreducible in Fq[T ]. In particular that number tends to infinity with q. MR Classification: 12E30 Directory: \Jarden\Diary\BSJ 23 December, 2011 * Alexander von Humboldt postdoc fellow. ** Research supported by the Minkowski Center for Geometry at Tel Aviv University, established by the Minerva Foundation, and by an ISF grant. Introduction Let f1, . . . , fr ∈ Z[X] be non-associate irreducible polynomials with positive leading coefficients. A conjecture of Bateman and Horn [BaH62, (1)] predicts for x > 1 that the number N(f1, . . . , fr;x) of positive integers 1 ≤ n ≤ x such that f1(n), . . . , fr(n) are prime numbers satisfies N(f1, . . . , fr;x) ∼ s(f1, . . . , fr) ∏r i=1 deg(fi) x log x ,
منابع مشابه
Prime Values of Polynomials and Irreducibility Testing
In 1857 Bouniakowsky [6] made a conjecture concerning prime values of polynomials that would, for instance, imply that x + 1 is prime for infinitely many integers x. Let ƒ (x) be a polynomial with integer coefficients and define the fixed divisor of ƒ, written d(ƒ), as the largest integer d such that d divides f(x) for all integers x. Bouniakowsky conjectured that if f(x) is nonconstant and irr...
متن کاملPrime Specialization in Higher Genus Ii
We continue the development of the theory of higher-genus Möbius periodicity that was studied in Part I for odd characteristic, now treating asymptotic questions and the case of characteristic 2. The extra difficulties in characteristic 2 are overcome via rigid geometry in characteristic 0. The results on Möbius periodicity in any positive characteristic are used to incorporate a correction fac...
متن کاملSimultaneous Prime Specializations of Polynomials over Finite Fields
Recently the author proposed a uniform analogue of the Bateman-Horn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq [T ] rather than Z[T ]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in particular ranges of the parameters. We give some applications including the solution of a proble...
متن کاملThe Auslander-Reiten Conjecture for Group Rings
This paper studies the vanishing of $Ext$ modules over group rings. Let $R$ be a commutative noetherian ring and $ga$ a group. We provide a criterion under which the vanishing of self extensions of a finitely generated $Rga$-module $M$ forces it to be projective. Using this result, it is shown that $Rga$ satisfies the Auslander-Reiten conjecture, whenever $R$ has finite global dimension and $ga...
متن کاملOn strongly J-clean rings associated with polynomial identity g(x) = 0
In this paper, we introduce the new notion of strongly J-clean rings associated with polynomial identity g(x) = 0, as a generalization of strongly J-clean rings. We denote strongly J-clean rings associated with polynomial identity g(x) = 0 by strongly g(x)-J-clean rings. Next, we investigate some properties of strongly g(x)-J-clean.
متن کامل