A note on Nakai’s conjecture for the ring $K[X₁,...,Xₙ]/(a₁X₁^{m} + ⋯ + aₙXₙ^{m})$

Paulo Roberto Brumatti; Marcelo Oliveira Veloso

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

On the generalized vanishing conjecture

Zhenzhen Feng, Xiaosong Sun (2019)

Czechoslovak Mathematical Journal

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We show that the GVC (generalized vanishing conjecture) holds for the differential operator $Λ = (\partial_{x} - Φ (\partial_{y})) \partial_{y}$ and all polynomials $P (x, y)$ , where $Φ (t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.

Results related to Huppert’s $ρ$ - $σ$ conjecture

Xia Xu, Yong Yang (2023)

Czechoslovak Mathematical Journal

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We improve a few results related to Huppert’s $ρ$ - $σ$ conjecture. We also generalize a result about the covering number of character degrees to arbitrary finite groups.

Greenberg’s conjecture for $ℤ_{p}^{d}$ -extensions

Andrea Bandini (2003)

Acta Arithmetica

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On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

A geometric construction for spectrally arbitrary sign pattern matrices and the $2 n$ -conjecture

Dipak Jadhav, Rajendra Deore (2023)

Czechoslovak Mathematical Journal

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We develop a geometric method for studying the spectral arbitrariness of a given sign pattern matrix. The method also provides a computational way of computing matrix realizations for a given characteristic polynomial. We also provide a partial answer to $2 n$ -conjecture. We determine that the $2 n$ -conjecture holds for the class of spectrally arbitrary patterns that have a column or row with at least $n - 1$ nonzero entries.

Semifields and a theorem of Abhyankar

Vítězslav Kala (2017)

Commentationes Mathematicae Universitatis Carolinae

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Abhyankar proved that every field of finite transcendence degree over $ℚ$ or over a finite field is a homomorphic image of a subring of the ring of polynomials $ℤ [T_{1}, \dots, T_{n}]$ (for some $n$ depending on the field). We conjecture that his result cannot be substantially strengthened and show that our conjecture implies a well-known conjecture on the additive idempotence of semifields that are finitely generated as semirings.

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 $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ 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 $\leq x$ 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 ( $n \geq 3$ ) real right-angled simplices. In this...

Non-Wieferich primes in number fields and $a b c$ -conjecture

Srinivas Kotyada, Subramani Muthukrishnan (2018)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be an algebraic number field of class number one and let $𝒪_{K}$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $𝒪_{K}$ under the assumption of the $a b c$ -conjecture for number fields.

On the Non-commutative Neutrix Product of $x_{+}^{l} a m b d a$ and $x_{+}^{- l a m b d a - r}$

Brian Fisher, Fatma Al-Sirehy (1999)

Publications de l'Institut Mathématique

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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 $f \in k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

A note on linear derivations

Amit Patra (2024)

Czechoslovak Mathematical Journal

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At first we prove some results on a general polynomial derivation using few results of linear derivation. Then we study the ring of constants of a linear derivation for some rings. We know that any linear derivation is a nonsimple derivation. In the last section we find the smallest integer $w > 1$ such that the polynomial ring in $n$ variables is $w$ -differentially simple, all $w$ derivations are nonsimple and the $w$ derivations set contains a linear derivation.

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 $q \geq 2$ and denote by $s_{q}$ the sum-of-digits function in base $q$ . For $j = 0, 1, \dots, q - 1$ consider $# {0 \leq n < N : s_{q} (2 n) \equiv j (mod q)} .$ 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 ${(\sum ℓ_{q})}_{p}$ ” (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 ${(\sum ℓ_{q})}_{p}$ ” [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 < ∞.

Rings in which elements are sum of a central element and an element in the Jacobson radical

Guanglin Ma, Yao Wang, André Leroy (2024)

Czechoslovak Mathematical Journal

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An element in a ring $R$ is called CJ if it is of the form $c + j$ , where $c$ belongs to the center and $j$ is an element from the Jacobson radical. A ring $R$ is called CJ if each element of $R$ is CJ. We establish the basic properties of CJ rings, give several characterizations of these rings, and connect this notion with many standard elementwise properties such as clean, uniquely clean, nil clean, CN, and CU. We study the behavior of this notion under various ring extensions. In particular, we show...

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 $σ_{A} (n ̅) \leq 5120$ for all n̅ ∈ ℤₘ.

Unimodular rows over Laurent polynomial rings

Abdessalem Mnif, Morou Amidou (2022)

Czechoslovak Mathematical Journal

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We prove that for any ring $𝐑$ of Krull dimension not greater than 1 and $n \geq 3$ , the group $E_{n} (𝐑 [X, X^{- 1}])$ acts transitively on ${Um}_{n} (𝐑 [X, X^{- 1}])$ . In particular, we obtain that for any ring $𝐑$ with Krull dimension not greater than 1, all finitely generated stably free modules over $𝐑 [X, X^{- 1}]$ are free. All the obtained results are proved constructively.

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 $1 \in k$ , $n \geq 2$ 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 $k$ .

On $^{γ (\cdot, \cdot)}$ -subdifferentiable and [+γ]-subdifferentiable Functions

S. Rolewicz (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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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 ) ”

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