Displaying similar documents to “Composite rational functions expressible with few terms”

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K . We prove Manin’s conjecture on the asymptotic (as T ) of the number of K -rational points of X of height less than T , and give an explicit construction of a measure on X ( 𝔸 ) , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points 𝐆 ( K ) on X ( 𝔸 ) . Our approach is based on the mixing property of L 2 ( 𝐆 ( K ) 𝐆 ( 𝔸 ) ) which we obtain with a rate of convergence. ...

Extensions of the Cugiani-Mahler theorem

Yann Bugeaud (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In 1955, Roth established that if ξ is an irrational number such that there are a positive real number ε and infinitely many rational numbers p / q with q 1 and | ξ - p / q | < q - 2 - ε , then ξ is transcendental. A few years later, Cugiani obtained the same conclusion with ε replaced by a function q ε ( q ) that decreases very slowly to zero, provided that the sequence of rational solutions to | ξ - p / q | < q - 2 - ε ( q ) is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it...

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

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Consider a recurrence sequence ( x k ) k of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a x k + 1 + a x k , where a , a , . . . , a n - 1 are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

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Let  X be a uniruled projective manifold and let  x be a general point. The main result of [2] says that if the ( - K X ) -degrees (i.e., the degrees with respect to the anti-canonical bundle of  X ) of all rational curves through x are at least dim X + 1 , then X is a projective space. In this paper, we study the structure of  X when the ( - K X ) -degrees of all rational curves through x are at least dim X . Our study uses the projective variety 𝒞 x T x ( X ) , called the VMRT at  x , defined as the union of tangent directions to the...

On the value set of small families of polynomials over a finite field, II

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

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We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in q [ T ] of degree d for which s consecutive coefficients a = ( a d - 1 , . . . , a d - s ) are fixed. Our estimate asserts that ( d , s , a ) = μ d q + ( q 1 / 2 ) , where μ d : = r = 1 d ( ( - 1 ) r - 1 ) / ( r ! ) . We also prove that ( d , s , a ) = μ ² d q ² + ( q 3 / 2 ) , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of q [ T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that ( d , 0 ) = μ ² d q ² + ( q ) , where ₂(d,0) denotes the average second moment for...

Equidistribution towards the Green current

Vincent Guedj (2003)

Bulletin de la Société Mathématique de France

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Let f : k k be a dominating rational mapping of first algebraic degree λ 2 . If S is a positive closed current of bidegree ( 1 , 1 ) on k with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks λ - n ( f n ) * S converge to the Green current T f . For some families of mappings, we get finer convergence results which allow us to characterize all f * -invariant currents.

Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

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Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension with respect to r . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime. ...

Obstruction sets and extensions of groups

Francesca Balestrieri (2016)

Acta Arithmetica

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Let X be a nice variety over a number field k. We characterise in pure “descent-type” terms some inequivalent obstruction sets refining the inclusion X ( k ) é t , B r X ( k ) B r . In the first part, we apply ideas from the proof of X ( k ) é t , B r = X ( k ) k by Skorobogatov and Demarche to new cases, by proving a comparison theorem for obstruction sets. In the second part, we show that if k are such that E x t ( , k ) , then X ( k ) = X ( k ) . This allows us to conclude, among other things, that X ( k ) é t , B r = X ( k ) k and X ( k ) S o l , B r = X ( k ) S o l k .

On τ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of τ -extending modules by τ -rational submodules and study some properties of such modules. It is shown that the set of all τ -rational left ideals of R R is a Gabriel filter. An R -module M is called τ -extending if every submodule of M is τ -rational in a direct summand of M . It is proved that M is τ -extending if and only if M = R e j M E ( R / τ ( R ) ) N , such that N is a τ -extending submodule of M . An example is given to show that the direct sum of τ -extending modules need not...

The supports of higher bifurcation currents

Romain Dujardin (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let ( f λ ) λ Λ be a holomorphic family of rational mappings of degree d on 1 ( ) , with k marked critical points c 1 , ... , c k . To this data is associated a closed positive current T 1 T k of bidegree ( k , k ) on Λ , aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c 1 , ... , c k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp ( T 1 T k ) . ...

Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1

(2016)

Acta Arithmetica

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For n ∈ ℕ, L > 0, and p ≥ 1 let κ p ( n , L ) be the largest possible value of k for which there is a polynomial P ≢ 0 of the form P ( x ) = j = 0 n a j x j , | a 0 | L ( j = 1 n | a j | p ) 1 / p , a j , such that ( x - 1 ) k divides P(x). For n ∈ ℕ, L > 0, and q ≥ 1 let μ q ( n , L ) be the smallest value of k for which there is a polynomial Q of degree k with complex coefficients such that | Q ( 0 ) | > 1 / L ( j = 1 n | Q ( j ) | q ) 1 / q . We find the size of κ p ( n , L ) and μ q ( n , L ) for all n ∈ ℕ, L > 0, and 1 ≤ p,q ≤ ∞. The result about μ ( n , L ) is due to Coppersmith and Rivlin, but our proof is completely different and much shorter even...

Homeomorphism groups of Sierpiński carpets and Erdős space

Jan J. Dijkstra, Dave Visser (2010)

Fundamenta Mathematicae

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Erdős space is the “rational” Hilbert space, that is, the set of vectors in ℓ² with all coordinates rational. Erdős proved that is one-dimensional and homeomorphic to its own square × , which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of . Let M n + 1 , n ∈ ℕ, be the n-dimensional Menger continuum in n + 1 , also known as the n-dimensional Sierpiński carpet, and let D be a countable dense subset of M n + 1 . We consider the topological group...