Equations in the Hadamard ring of rational functions

Ferretti, Andrea; Zannier, Umberto

doi:10.2422/2036-2145.2007.3.05

Cited by 7 publications

(5 citation statements)

References 8 publications

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“…The first three cases reduce to the WIHP for G 2 m . This is already established and there are much more general results regarding algebraic groups (see [4,6,12,25]). The most difficult case is the one of the smooth cubic.…”

Section: The Wihp For the Complement Of The Smooth Plane Cubicmentioning

confidence: 81%

The Hilbert Property for integral points of affine smooth cubic surfaces

Coccia

2019

Journal of Number Theory

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In this paper we prove that the set of S-integral points of the smooth cubic surfaces in A 3 over a number field k is not thin, for suitable k and S. As a corollary, we obtain results on the complement in P 2 of a smooth cubic curve, improving on Beukers' proof that the S-integral points are Zariski dense, for suitable S and k. With our method we reprove Zariski density, but our result is more powerful since it is a stronger form of Zariski density. We moreover prove that the rational integer points on the Fermat cubic surface x 3 + y 3 + z 3 = 1 form a non-thin set and we link our methods to previous results of Lehmer, Miller-Woollett and Mordell. 1 2 SIMONE COCCIA Theorem 1.5 ([9]). The K3 surface defined in P 3 by x 4 + y 4 = z 4 + w 4is not (uni)rational over C and has the Hilbert property over Q.In the recent preprint [10] J. L. Demeio provided further examples of varieties with the HP, such as K3 surfaces or quotients of varieties under the action of a finite group. Corvaja and Zannier formulated analogous definitions and conjectures for the integer case, which are the object of this paper.Definition 1.6. We say that an algebraic variety X/k has the S-integral Hilbert propertyEquivalently, X has the IHP if given finitely many k-covers π i : Y i → X, each without rational sections defined overk, the setRemark 1.7. One typical assumption is that the covers are given by finite maps π : Y → X where Y is a normal absolutely irreducible variety. In fact, consider a cover π : Y → X where Y is an absolutely irreducible variety. By removing a closed set, we can assume that π is a dominant morphism. We can mutually decompose Y and X in affine open sets such that π restricts to π : V → U . The map π is still a dominant morphism and so it defines an inclusion π * :one obtains a normal absolutely irreducible affine varietyṼ = Spec(A) equipped with a finite morphismπ :Ṽ → U and a birational isomorphism f : V Ṽ such that π =π • f . Gluing these affine spectra we obtain the desired finite map, which differs from π only on a closed set, thus proving that this reduction is possible. The original version of Hilbert's Irreducibility theorem then becomes:Theorem 1.8. The affine line A 1 has the IHP over Z.The natural analogue of Problem 1.4 is then: Problem 1.9. Any smooth quasi-projective variety topologically simply connected and with a Zariski-dense set of S-integral points has the IHP.In order to ask a similar question for non-simply connected varieties, one is led to the following definition.Definition 1.10 (Weak integral Hilbert property). A normal quasi-projective algebraic variety X/k has the weak integral Hilbert property (WIHP) if, given finitely many covers π i : Y i → X each ramified above a non-empty divisor, the set X(O S ) \ π i (Y i (k)) is Zariski dense. Problem 1.11. If a normal quasi-projective algebraic variety has a Zariski dense set of S-integral points then it has the WIHP.Remark 1.12. Clearly for simply connected varieties the IHP is equivalent to the WIHP.Remark 1.13. It could be possible that we should...

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Section: The Wihp For the Complement Of The Smooth Plane Cubicmentioning

confidence: 81%

The Hilbert Property for integral points of affine smooth cubic surfaces

Coccia

2019

Journal of Number Theory

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“…the set of sequences in K satisfying a linear recurrence relation. Let us first state the following version from [8]:…”

Section: Preliminariesmentioning

confidence: 99%

“…Furthermore, note that for any z n we can find a finite extension of K which contains this element z n . Hence we get a bound (8) h…”

mentioning

confidence: 96%

Integral zeros of a polynomial with linear recurrences as coefficients

Fuchs¹,

Heintze²

2020

Preprint

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Let K be a number field, S a finite set of places of K, and O S be the ring of S-integers. Moreover, letbe a polynomial in Z having simple linear recurrences of integers evaluated at n as coefficients. Assuming some technical conditions we give a description of the zeros (n, z) ∈ N × O S of the above polynomial. We also give a result in the spirit of Hilbert irreducibility for such polynomials.

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“…We can naturally formulate also an analogue of Question-Conjecture 2. For brevity we do not repeat this, and only recall that some results in this direction are proved in [9], [14], [41] for multiplicative tori X = G n m . In this toric case the S-integral points are those with S-unit coordinates, and they form a finitely generated group.…”

Section: 2mentioning

confidence: 99%

“…This motivates the definition of a new property, including in a sense the HP, and seemingly with much the same applications as the HP. See §2 for further comments, especially §2.2 for an explicit statement, and see the papers [9], [14], [41] for some results regarding implicitly this different property.…”

Section: Introductionmentioning

confidence: 99%

On the Hilbert property and the fundamental group of algebraic varieties

Corvaja

Zannier

2016

Math. Z.

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In this paper we link the so-called Hilbert property (HP) for an algebraic variety (over a number field) with its fundamental group, in a perspective which appears new. (The notion of HP comes from Hilbert’s Irreducibility Theorem and has important implications, for instance towards the Inverse Galois Problem.) We shall observe that the HP is in a sense ‘opposite’ to the Chevalley–Weil Theorem. This shall immediately entail the result that the HP can possibly hold only for simply connected varieties (in the appropriate sense). In turn, this leads to new counterexamples to the HP, involving Enriques surfaces. In this view, we shall also formulate an alternative related property, possibly holding in full generality, and some conjectures which unify part of what is known in the topic. These predict that for a variety with a Zariski-dense set of rational points, the validity of the HP, which is defined arithmetically, is indeed of purely topological nature. Also, a consequence of these conjectures would be a positive solution of the Inverse Galois Problem. In the paper we shall also prove the HP for a K3 surface related to the above Enriques surface, providing what appears to be the first example of a variety non rational (over (Formula presented.)) for which the HP can be proved. In an Appendix we shall further discuss, among other things, the HP also for Kummer surfaces. All of this basically exhausts the study of the HP for surfaces with a Zariski-dense set of rational points

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Equations in the Hadamard ring of rational functions

Cited by 7 publications

References 8 publications

The Hilbert Property for integral points of affine smooth cubic surfaces

The Hilbert Property for integral points of affine smooth cubic surfaces

Integral zeros of a polynomial with linear recurrences as coefficients

On the Hilbert property and the fundamental group of algebraic varieties

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