Schinzel's Hypothesis Research Articles

Schinzel Hypothesis on average and rational points

We resolve Schinzel’s Hypothesis (H) for 100% of polynomials of arbitrary degrees. We deduce that a positive proportion of diagonal conic bundles over {mathbb {Q}} with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations.

Common divisors of totients of polynomial sequences

Motivated by a question of Venkataramana, we consider the greatest common divisor of $\phi(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we establish that this gcd may be bounded just in terms of the degree of the polynomial $f$. Unconditionally we establish such a bound for quadratic polynomials, as well as polynomials that split completely into linear factors. The paper also addresses a question of Calegari, and establishes that there are infinitely many $n$ such that $n^2+1$ is not divisible by any prime $\equiv 1 \bmod 2^m$ provided $m$ is a large fixed integer.

Integral points on generalised affine Châtelet surfaces

We show, conditionally on Schinzel's hypothesis, that the only obstruction to the integral Hasse principle for a large subfamily of generalised affine Châtelet surfaces is the Brauer–Manin one.

Rational points and prime values of polynomials in moderately many variables

We derive the Hasse principle and weak approximation for fibrations of certain varieties in the spirit of work by Colliot-Thélène–Sansuc and Harpaz–Skorobogatov–Wittenberg. Our varieties are defined through polynomials in many variables and part of our work is devoted to establishing Schinzel's hypothesis for polynomials of this kind. This last part is achieved by using arguments behind Birch's well-known result regarding the Hasse principle for complete intersections with the notable difference that we prove our result in 50% fewer variables than in the classical Birch setting. We also study the problem of square-free values of an integer polynomial with 66.6% fewer variables than in the Birch setting.

On the number of certain del Pezzo surfaces of degree four violating the Hasse principle

We give an asymptotic expansion for the density of del Pezzo surfaces of degree four in a certain Birch Swinnerton-Dyer family violating the Hasse principle due to a Brauer–Manin obstruction. Under the assumption of Schinzel's hypothesis and the finiteness of Tate–Shafarevich groups for elliptic curves, we obtain an asymptotic formula for the number of all del Pezzo surfaces in the family, which violate the Hasse principle.

Principes locaux-globaux pour certaines fibrations en torseurs sous un tore

AbstractLetkbe a number field andTak-torus. Consider a family of torsors underT, i.e. a morphismf:X→ ℙ1kfrom a projective, smoothk-varietyXto ℙ1k, the generic fibreXη→ η of which is a smooth compactification of a principal homogeneous space underT⊗kη. We study the Brauer–Manin obstruction to the Hasse principle and to weak approximation forX, assuming Schinzel's hypothesis. We generalise Wei's recent results [21]. Our results are unconditional ifk=Qand all non-split fibres offare defined overQ. We also establish an unconditional analogue of our main result for zero-cycles of degree 1.

Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups

We show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical for some projection away from a plane g:X⤏P1, i.e., that every Brauer class is obtained by pullback from an element of Brk(P1). As a consequence, we prove that a Brauer class obstructs the existence of a k-rational point if and only if all k-fibers of g fail to be locally solvable, or in other words, if and only if X is covered by curves that each have no adelic points. Using work of Wittenberg, we deduce that for certain quartic del Pezzo surfaces with nontrivial Brauer group the algebraic Brauer–Manin obstruction is sufficient to explain all failures of the Hasse principle, conditional on Schinzel's hypothesis and the finiteness of Tate–Shafarevich groups. The proof of the main theorem is constructive and gives a simple and practical algorithm, distinct from that in [5], for computing all classes in the Brauer group of X (modulo constant algebras).

Local-global principle for certain biquadratic normic bundles

Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})/k}({x})=Q(t_{1},...,t_{m})^{2}$ over a number field $k$. We prove that : (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of $X$; (2) the analogue for rational points is also valid assuming Schinzel's hypothesis.

Integral Brauer-Manin obstructions for sums of two squares and a power

We use Brauer–Manin obstructions to explain failures of the integral Hasse principle and strong approximation away from ∞ for the equation x2+y2+zk=m with fixed integers k⩾3 and m. Under Schinzel's hypothesis (H), we prove that Brauer–Manin obstructions corresponding to specific Azumaya algebras explain all failures of strong approximation away from ∞ at the variable z. Finally, we present an algorithm that, again under Schinzel's hypothesis (H), finds out whether the equation has any integral solutions.

Higher-dimensional analogs of Châtelet surfaces

We discuss the geometry and arithmetic of higher-dimensional analogs of Châtelet surfaces; namely, we describe the structure of their Brauer and Picard groups and show that they can violate the Hasse principle. In addition, we use these varieties to give straightforward generalizations of two recent results of Poonen. Specifically, we prove that, assuming Schinzel's hypothesis, the non-mth powers of a number field are diophantine. Also, given a global field k such that Char(k)=p or k contains the pth roots of unity, we construct a (p+1)-fold that has no k-points and no étale-Brauer obstruction to the Hasse principle.

Irreducible values of polynomials

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove a quantitative arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields via model theory. A main tool in the proof is an irreducibility theorem à la Hilbert.

On a conjecture of Beard, O'Connell and West concerning perfect polynomials

Following Beard, O'Connell and West [J.T.B. Beard Jr., J.R. O'Connell Jr., K.I. West, Perfect polynomials over GF ( q ) , Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 62 (1977) 283–291] we call a polynomial over a finite field F q perfect if it coincides with the sum of its monic divisors. The study of perfect polynomials was initiated in 1941 by Carlitz's doctoral student Canaday in the case q = 2 , who proposed the still unresolved conjecture that every perfect polynomial over F 2 has a root in F 2 . Beard et al. later proposed the analogous hypothesis for all finite fields. Counterexamples to this general conjecture were found by Link (in the cases q = 11 , 17 ) and Gallardo & Rahavandrainy (in the case q = 4 ). Here we show that the Beard–O'Connell–West conjecture fails in all cases except possibly when q is prime. When q = p is prime, utilizing a construction of Link we exhibit a counterexample whenever p ≡ 11 or 17 ( mod 24 ) . On the basis of a polynomial analog of Schinzel's Hypothesis H, we argue that if there is a single perfect polynomial over the finite field F q with no linear factor, then there are infinitely many. Lastly, we prove without any hypothesis that there are infinitely many perfect polynomials over F 11 with no linear factor.

Principe de Hasse pour les intersections de deux quadriques

Assuming Schinzel's hypothesis and the finiteness of Tate–Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics in n-dimensional projective space satisfy the Hasse principle if n ⩾ 5 . The same result holds for n = 4 , i.e., for del Pezzo surfaces of degree 4, provided the Brauer group is reduced to constants and the surface is sufficiently general. Detailed proofs of the results announced herein will be published later on. To cite this article: O. Wittenberg, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

2-Descent on elliptic curves and rational points on certain Kummer surfaces

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579–650]. To describe the rest, let E ( 1 ) and E ( 2 ) be elliptic curves, D ( 1 ) and D ( 2 ) their respective 2-coverings, and X be the Kummer surface attached to D ( 1 ) × D ( 2 ) . In the appendix we study the Brauer–Manin obstruction to the existence of rational points on X . In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer–Manin obstructions. This second part depends on the hypothesis that the relevent Tate–Shafarevich group is finite, but it does not require Schinzel's Hypothesis.

More on an undecidability result of Bateman, Jockusch and Woods

Let P be the set of prime numbers. Theorem 1 of [1] shows that the linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,P〉 and therefore that the first-order theory of this structure is undecidable. Let m be any fixed natural number >2, let R be the set of natural numbers <m which are prime to m, and let r be any fixed element of R. The setis infinite (Dirichlet). Theorem 1 of [1] can be improved as follows:Proposition. The linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,Pm,r〉 and therefore that the first-order theory of this structure is undecidable.Proof. We follow [1] with the following new ingredients. Let k be the number of elements of R, i.e. k = ϕ(m) where ϕ is Euler's totient function. Since k is even, the polynomial g(n) = nk + n satisfies g(0) = g(−1) = 0, so (by Lemma 1 of [1]) it follows from the linear case of (H) that there are natural numbers al (l ϵ ω) such that al+g(0), al+g(1),…, al+g(l) are consecutive primes. Since R is finite, we may assume that all the al's have the same residue t in R, so that al+g(i) ≡ t+1+i (mod m) for i ϵ R. This implies that the function t+1+i (reduced mod m) gives a permutation of R, so we can find s ϵ R such that al+g(s) ≡ r (mod m). Consider the polynomial h(n) = g(s + mn) and let bl = as+ml. Then bl + h(0), bl + h(1),…, bl + h(l) are elements of Pm,r. They are not necessarily consecutive elements of Pm,r, but they are separated by a fixed number of elements of Pm,r. This implies that {h(n) ∣ n ϵ ω} is definable in 〈ω,+,Pm,r〉(by adapting the proof of Theorem 1 of [1]), and the result follows.

Rational points and zero-cycles on fibred varieties: Schinzel's hypothesis and Salberger's device

Rational points and zero-cycles on fibred varieties: Schinzel's hypothesis and Salberger's device

Decidability and undecidability of theories with a predicate for the primes

AbstractIt is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.

On Sierpinski's Conjecture Concerning the Euler Totient

AbstractIf Φk(n) denotes the Schemmel totient (so that Φ1 (n) becomes the Euler totient) we conjecture that for each k ≥ 1 and any given integer n > 1 there exist infinitely many m for which the equation Φk(x) = m has exactly n solutions. For the case k = 1, this gives Sierpinski's conjecture.We prove that on the basis of Schinzel's Hypothesis H, our conjecture holds for any k ≥ 3 of the form where p0 is an odd prime and α ∊ N. In 1961 Schinzel proved the case k = 1 assuming his Hypothesis H.