A note on the density of <i>k</i>-free polynomial sets, Haar measure and global fields

Demangos, L.; Longhi, Ignazio

doi:10.2989/16073606.2021.1945701

Cited by 2 publications

(7 citation statements)

References 16 publications

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“…4.2.1 we will give a working definition of a density on D n , but readers should be aware that our goal is simply to isolate the properties it should satisfy for comparison with the Haar measure on D n . A number of examples of such densities will be discussed in full detail in the companion paper [18]; in the present work, Sects. 4.3.1-4.3.3, we just quickly touch upon some cases that will be needed later.…”

Section: Structure Of This Papermentioning

confidence: 99%

“…The easy part of proving ( 18 The claim Irr1 is obvious: if Irr(p) is not empty then it is a D * -orbit and (17) shows that D * Irr(D) is the union of the closures of these orbits.…”

Section: Theorem 32 If D Is a Global Dedekind Domain Thenmentioning

confidence: 99%

“…The difference is (33), which (as proved in [32, Example 7]) is not a consequence of those axioms: we postulate it in order to have equality (34) below, which will play a crucial role in the proof of Theorem 4.19. For a more precise comparison between our conditions and [32], see [18].…”

Section: Definition Of Densitymentioning

confidence: 99%

“…4.2.1. For more (including generalizations to global Dedekind domains of the most commonly used densities over N) see [18]. (30) show that the equality…”

Section: Examples Of Densitiesmentioning

confidence: 99%

“…enjoys the desired properties. We refer to [18] for a detailed proof, as well as for alternative definitions of asymptotic densities. (Some special cases can be found in [17,Proposition 2.11].…”

Section: Asymptotic Densitiesmentioning

confidence: 99%

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Densities on Dedekind domains, completions and Haar measure

Demangos,

Longhi

2024

Math. Z.

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Let D be the ring of S-integers in a global field and D its profinite completion. Given X ⊆ D n , we consider its closure X ⊆ D n and ask what can be learned from X about the "size" of X .In particular, we ask when the density of X is equal to the Haar measure of X . We provide a general definition of density which encompasses the most commonly used ones. Using it we give a necessary and sufficient condition for the equality between density and measure which subsumes a criterion due to Poonen and Stoll. We also show how Ekedahl's sieve fits into our setting and find conditions ensuring that X can be written as a product of local closures. In another direction, we extend the Davenport-Erdős theorem to every D as above and offer a new interpretation of it as a "density=measure" result. Our point of view also provides a simple proof that in any D the set of elements divisible by at most k distinct primes has density 0 for any k ∈ N. Finally, we show that the closure of the set of prime elements of D is the union of the group of units of D with a negligible part.

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Section: Structure Of This Papermentioning

confidence: 99%

“…The easy part of proving ( 18 The claim Irr1 is obvious: if Irr(p) is not empty then it is a D * -orbit and (17) shows that D * Irr(D) is the union of the closures of these orbits.…”

Section: Theorem 32 If D Is a Global Dedekind Domain Thenmentioning

confidence: 99%

Section: Definition Of Densitymentioning

confidence: 99%

“…4.2.1. For more (including generalizations to global Dedekind domains of the most commonly used densities over N) see [18]. (30) show that the equality…”

Section: Examples Of Densitiesmentioning

confidence: 99%

Section: Asymptotic Densitiesmentioning

confidence: 99%

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Densities on Dedekind domains, completions and Haar measure

Demangos,

Longhi

2024

Math. Z.

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Quadratic forms and Genus Theory: A link with 2‐descent and an application to nontrivial specializations of ideal classes

Dallaporta

2024

Journal of London Math Soc

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Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well‐known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) . When , we show that the Genus Theory map is the quadratic form version of the 2‐descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.

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A note on the density of k-free polynomial sets, Haar measure and global fields

Cited by 2 publications

References 16 publications

Densities on Dedekind domains, completions and Haar measure

Densities on Dedekind domains, completions and Haar measure

Quadratic forms and Genus Theory: A link with 2‐descent and an application to nontrivial specializations of ideal classes

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