Displaying similar documents to “A note on Nakai’s conjecture for the ring K [ X ₁ , . . . , X ₙ ] / ( a ₁ X ₁ m + ⋯ + a ₙ X ₙ m ) ”

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let p i denote the ith prime. We conjecture that there are precisely 28 solutions to the equation n ² - 1 = p ₁ α ₁ ⋯ p k α k in positive integers n and α₁,..., α k . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function ψ ( x , y ) which is the number of positive integers ≤ x and free of prime factors > y . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( n ≥ 3 ) real right-angled simplices. In this...

Recent progress on the Jacobian Conjecture

Michiel de Bondt, Arno van den Essen (2005)

Annales Polonici Mathematici

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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form x + ( A x ) * 3 with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with f ∈ k [ n ] homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let q ≥ 2 and denote by s q the sum-of-digits function in base q . For j = 0 , 1 , ⋯ , q - 1 consider # { 0 ≤ n < N : s q ( 2 n ) ≡ j ( mod q ) } . In 1983, F. M. Dekking conjectured that this quantity is greater than N / q and, respectively, less than N / q for infinitely many N , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

Corrigendum to “Commutators on ( ∑ ℓ q ) p ” (Studia Math. 206 (2011), 175-190)

Dongyang Chen, William B. Johnson, Bentuo Zheng (2014)

Studia Mathematica

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We give a corrected proof of Theorem 2.10 in our paper “Commutators on ( ∑ ℓ q ) p ” [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

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Given a set A ⊂ ℕ let σ A ( n ) denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, σ A ( n ) is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and σ A ( n ̅ ) ≤ 5120 for all n̅ ∈ ℤₘ.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with p g = 0 obtained as 0 -sets X σ of a section σ of a very ample vector bundle on a variety X with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as 0 on holomorphic 2 -forms of  X σ , then it acts as 0 on  0 -cycles of degree 0 of  X σ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general X σ ...

On matrix Lie rings over a commutative ring that contain the special linear Lie ring

Evgenii L. Bashkirov, Esra Pekönür (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let K be an associative and commutative ring with 1 , k a subring of K such that 1 ∈ k , n ≥ 2 an integer. The paper describes subrings of the general linear Lie ring g l n ( K ) that contain the Lie ring of all traceless matrices over k .

Selection principles and upper semicontinuous functions

Masami Sakai (2009)

Colloquium Mathematicae

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In connection with a conjecture of Scheepers, Bukovský introduced properties wQN* and SSP* and asked whether wQN* implies SSP*. We prove it in this paper. We also give characterizations of properties S₁(Γ,Ω) and S f i n ( Γ , Ω ) in terms of upper semicontinuous functions

Eigenspaces of the ideal class group

Cornelius Greither, Radan Kučera (2014)

Annales de l’institut Fourier

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The aim of this paper is to prove an analog of Gras’ conjecture for an abelian field F and an odd prime p dividing the degree [ F : ℚ ] assuming that the p -part of Gal ( F / ℚ ) group is cyclic.

Equations in the Hadamard ring of rational functions

Andrea Ferretti, Umberto Zannier (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let K be a number field. It is well known that the set of recurrencesequences with entries in K is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume { a n } is a recurrence sequence and suppose that all the a n have a d th root in the field...

The Jacobian Conjecture in case of "non-negative coefficients"

Ludwik M. Drużkowski (1997)

Annales Polonici Mathematici

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It is known that it is sufficient to consider in the Jacobian Conjecture only polynomial mappings of the form F ( x ₁ , . . . , x n ) = x - H ( x ) : = ( x ₁ - H ₁ ( x ₁ , . . . , x n ) , . . . , x n - H n ( x ₁ , . . . , x n ) ) , where H j are homogeneous polynomials of degree 3 with real coefficients (or H j = 0 ), j = 1,...,n and H’(x) is a nilpotent matrix for each x = ( x ₁ , . . . , x n ) ∈ ℝ n . We give another proof of Yu’s theorem that in the case of non-negative coefficients of H the mapping F is a polynomial automorphism, and we moreover prove that in that case d e g F - 1 ≤ ( d e g F ) i n d F - 1 , where i n d F : = m a x i n d H ' ( x ) : x ∈ ℝ n . Note that the above inequality is not true when the coefficients...

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

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Let G be a finite group, and let N ( G ) be the set of conjugacy class sizes of G . By Thompson’s conjecture, if L is a finite non-abelian simple group, G is a finite group with a trivial center, and N ( G ) = N ( L ) , then L and G are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

Separating by G δ -sets in finite powers of ω₁

Yasushi Hirata, Nobuyuki Kemoto (2003)

Fundamenta Mathematicae

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It is known that all subspaces of ω₁² have the property that every pair of disjoint closed sets can be separated by disjoint G δ -sets (see [4]). It has been conjectured that all subspaces of ω₁ⁿ also have this property for each n < ω. We exhibit a subspace of ⟨α,β,γ⟩ ∈ ω₁³: α ≤ β ≤ γ which does not have this property, thus disproving the conjecture. On the other hand, we prove that all subspaces of ⟨α,β,γ⟩ ∈ ω₁³: α < β < γ have this property.

On some noetherian rings of C ∞ germs on a real closed field

Abdelhafed Elkhadiri (2011)

Annales Polonici Mathematici

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Let R be a real closed field, and denote by R , n the ring of germs, at the origin of Rⁿ, of C ∞ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring R , n ⊂ R , n with some natural properties. We prove that, for each n ∈ ℕ, R , n is a noetherian ring and if R = ℝ (the field of real numbers), then ℝ , n = ₙ , where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring R , n . ...

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

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The Littlewood conjecture in Diophantine approximation claims that inf q ≥ 1 q · ∥ q α ∥ · ∥ q β ∥ = 0 holds for all real numbers α and β , where ∥ · ∥ denotes the distance to the nearest integer. Its p -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that inf q ≥ 1 q · ∥ q α ∥ · | q | p = 0 holds for every real number α and every prime number p , where | · | p denotes the p -adic absolute value normalized by | p | p = p - 1 . We survey the known results on these conjectures and highlight recent developments. ...