Problems and results on the distribution of algebraic points on algebraic varieties

Bombieri, Enrico

doi:10.5802/jtnb.656

Cited by 8 publications

(25 citation statements)

References 17 publications

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“…The equivalent family above would also imply similar bounds for odd numbers of variables, e.g. an upper bound for 2s = 5 (a problem raised by [Bom09]). At first glance, there does not appear to be an equivalent Diophantine problem involving 4 variables, or any odd number 2s, although the general "restricted arcs" form of the 4 variable problem (s = 2) could in principle pare away most, if not all, but the most extreme minor arcs for s ≥ 3.…”

Section: Contribution From Singular Hyperplane Sectionsmentioning

confidence: 98%

Diagonal cubic forms and the large sieve

Wang¹

2021

Preprint

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Let F (x) be a diagonal integer-coefficient cubic form in m ∈ {4, 5, 6} variables. Excluding rational lines if m = 4, we bound the number of integral solutions x ∈ [−X, X] m to F (x) = 0 by O F, (X 3m/4−3/2+ ), conditionally on an "optimal large sieve inequality" (in a specific range of parameters) for "approximate Hasse-Weil L-functions" of smooth hyperplane sectionsWhen m is even, these results were previously established conditionally under Hooley's Hypothesis HW. Our 2 large sieve approach requires that certain bad factors be roughly 1 on average in 2 , while the ∞ Hypothesis HW approach only required the bound in 1 . Furthermore, the large sieve only accepts uniform vectors; yet our "initially given" vectors are only approximately uniform over c, due to variation in bad factors and in the archimedean component. Nonetheless, after some bookkeeping, partial summation, and Cauchy, the large sieve will still apply.

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Section: Contribution From Singular Hyperplane Sectionsmentioning

confidence: 98%

Diagonal cubic forms and the large sieve

Wang¹

2021

Preprint

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“…Here, the implications thus far discovered tend to go with Weyl's direction, from number theory to dynamics; and here too, the Diophantine questions were formulated and studied, for their intrinsic interest, long before their dynamical significance could be found. The celebrated Lehmer problem, on the spectral gap in the Mahler measure (see Bombieri [15] for a perspective in Diophantine Geometry), was shown by Lind to be equivalent to either of the following dynamical questions: [59] does the infinite torus (R/Z) Z admit an ergodic automorphism with finite entropy? [60] are all Bernoulli shifts algebraizable, in the form of a measurable equivalence with an automorphism of a compact group?…”

Section: Diophantine/dynamical Pairsmentioning

confidence: 99%

Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian $\mathbb{Z}^d$-actions

Dimitrov

2016

Preprint

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We prove a sub-Liouville bound, up to a uniformly bounded exceptional set, on the distance from an N -torsion point of G d m/Q to an algebraic subset, under a fixed Archimedean place Q ֒→ C. As a consequence, for all non-zero integer Laurent polynomials P in d commuting variables, we prove that the averages of logBy the work of B. Kitchens, D. Lind, K. Schmidt and T. Ward, this convergence consequence amounts to the following statement in dynamics: For every Noetherian Z d -action T : Z d → Aut(X) by automorphisms of a compact abelian group X having a finite topological entropy h(T ), the annihilator Per N (T ) of N • Z d has e (1+o(1))h(T )N d connected components, as N → ∞. Moreover, it follows that all weak- * limit measures of the push-forwards of the Haar measures on Per N (T ), under any a sequence of positive integers N , are measures of maximum entropy h(T ).Combined with work of Thang Le, this solves the abelian case of the problem of the asymptotic growth of torsion in the homology of congruence covers of a fixed finite simplicial complex. We also give an indication on how an alternative, completely different route to the convergence result and its consequences is also possible through Habegger's recent work on Diophantine approximation to definable sets.

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“…Problem 1 (Bombieri, 2009). Determine workable conditions for the validity of the Northcott property for subfields of Q.…”

Section: Definition 1 (Northcott Property)mentioning

confidence: 99%

On certain infinite extensions of the rationals with Northcott property

Widmer

2009

Monatsh Math

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A set of algebraic numbers has the Northcott property if each of its subsets of bounded Weil height is finite. Northcott's Theorem, which has many Diophantine applications, states that sets of bounded degree have the Northcott property. Bombieri, Dvornicich and Zannier raised the problem of finding fields of infinite degree with this property. Bombieri and Zannier have shown that $\IQ_{ab}^{(d)}$, the maximal abelian subfield of the field generated by all algebraic numbers of degree at most $d$, is such a field. In this note we give a simple criterion for the Northcott property and, as an application, we deduce several new examples, e.g. $\IQ(2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...)$ has the Northcott property if and only if $2^{1/d_1},3^{1/d_2},5^{1/d_3},7^{1/d_4},11^{1/d_5},...$ tends to infinity

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Problems and results on the distribution of algebraic points on algebraic varieties

Cited by 8 publications

References 17 publications

Diagonal cubic forms and the large sieve

Diagonal cubic forms and the large sieve

Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian $\mathbb{Z}^d$-actions

On certain infinite extensions of the rationals with Northcott property

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