On the existence of dimension zero divisors in algebraic function fields defined over $_q$

S. Ballet; C. Ritzenthaler; R. Rolland

Displaying similar documents to “On the existence of dimension zero divisors in algebraic function fields defined over $_{q}$ ”

On reduced Arakelov divisors of real quadratic fields

Ha Thanh Nguyen Tran (2016)

Acta Arithmetica

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We generalize the concept of reduced Arakelov divisors and define C-reduced divisors for a given number C ≥ 1. These C-reduced divisors have remarkable properties, similar to the properties of reduced ones. We describe an algorithm to test whether an Arakelov divisor of a real quadratic field F is C-reduced in time polynomial in $l o g | Δ_{F} |$ with $Δ_{F}$ the discriminant of F. Moreover, we give an example of a cubic field for which our algorithm does not work.

Stein open subsets with analytic complements in compact complex spaces

Jing Zhang (2015)

Annales Polonici Mathematici

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Let Y be an open subset of a reduced compact complex space X such that X - Y is the support of an effective divisor D. If X is a surface and D is an effective Weil divisor, we give sufficient conditions so that Y is Stein. If X is of pure dimension d ≥ 1 and X - Y is the support of an effective Cartier divisor D, we show that Y is Stein if Y contains no compact curves, $H^{i} (Y,_{Y}) = 0$ for all i > 0, and for every point x₀ ∈ X-Y there is an n ∈ ℕ such that $Φ_{| n D |}^{- 1} (Φ_{| n D |} (x ₀)) \cap Y$ is empty or has dimension 0, where $Φ_{| n D |}$ ...

The existence of solutions of the generalized pseudoprime congruence $a^{f (n)} \equiv b^{f (n)} (m o d n)$

Wayne L. McDaniel (1990)

Colloquium Mathematicae

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On a linear homogeneous congruence

A. Schinzel, M. Zakarczemny (2006)

Colloquium Mathematicae

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The number of solutions of the congruence $a ₁ x ₁ + \dots + a_{k} x_{k} \equiv 0 (m o d n)$ in the box $0 \leq x_{i} \leq b_{i}$ is estimated from below in the best possible way, provided for all i,j either $(a_{i}, n) | (a_{j}, n)$ or $(a_{j}, n) | (a_{i}, n)$ or $n | [a_{i}, a_{j}]$ .

Subfields of henselian valued fields

Ramneek Khassa, Sudesh K. Khanduja (2010)

Colloquium Mathematicae

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Let (K,v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and $v_{k}$ be the valuation obtained by restricting v to k. We give some necessary and sufficient conditions for $(k, v_{k})$ to be henselian. In particular, it is shown that if k is dense in its henselization, then $(k, v_{k})$ is henselian. We deduce some well known results proved in this direction through other considerations.

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

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Let $ℚ (α)$ and $ℚ (β)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $ℚ (β) \to ℚ (α)$ . The algorithm is particularly efficient if there is only one isomorphism.

A $C^{*}$ -Algebra Approach to Field Theory

D. Kastler (1968)

Recherche Coopérative sur Programme n°25

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On divisibility of $h^{+}$ by the prime $5$

Stanislav Jakubec (1994)

Mathematica Slovaca

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Sum of higher divisor function with prime summands

Yuchen Ding, Guang-Liang Zhou (2023)

Czechoslovak Mathematical Journal

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Let $l ⩾ 2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $\sum_{1 ⩽ n_{1}, n_{2}, ..., n_{l} ⩽ x^{1 / 2}} τ_{k} (n_{1}^{2} + n_{2}^{2} + \dots + n_{l}^{2}),$ where $τ_{k} (n)$ represents the $k$ th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum $\sum_{1 ⩽ p_{1}, p_{2}, ..., p_{l} ⩽ x} τ_{k} (p_{1} + p_{2} + \dots + p_{l}),$ where $p_{1}, p_{2}, \dots, p_{l}$ are prime variables.

Hyperbolicity and integral points off divisors in subgeneral position in projective algebraic varieties

Do Duc Thai, Nguyen Huu Kien (2015)

Acta Arithmetica

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The purpose of this article is twofold. The first is to find the dimension of the set of integral points off divisors in subgeneral position in a projective algebraic variety $V \subset ℙ_{k ̅}^{m}$ , where k is a number field. As consequences, the results of Ru-Wong (1991), Ru (1993), Noguchi-Winkelmann (2003) and Levin (2008) are recovered. The second is to show the complete hyperbolicity of the complement of divisors in subgeneral position in a projective algebraic variety $V \subset ℙ_{ℂ}^{m} .$

A formula for the number of solutions of a restricted linear congruence

K. Vishnu Namboothiri (2021)

Mathematica Bohemica

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Consider the linear congruence equation $x_{1} + ... + x_{k} \equiv b (mod n^{s})$ for $b \in ℤ$ , $n, s \in ℕ$ . Let ${(a, b)}_{s}$ denote the generalized gcd of $a$ and $b$ which is the largest $l^{s}$ with $l \in ℕ$ dividing $a$ and $b$ simultaneously. Let $d_{1}, ..., d_{τ (n)}$ be all positive divisors of $n$ . For each $d_{j} ∣ n$ , define $𝒞_{j, s} (n) = {1 \leq x \leq n^{s} : {(x, n^{s})}_{s} = d_{j}^{s}}$ . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on $x_{i}$ . We generalize their result with generalized gcd restrictions on $x_{i}$ and prove that for the above linear congruence, the...

Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca, Edoardo Sernesi (2014)

Journal de l’École polytechnique — Mathématiques

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We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve $C$ and the stability of the sheaf of logarithmic vector fields along $C$ , the freeness of the divisor $C$ and the Torelli properties of $C$ (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

On congruence permutable $G$ -sets

Attila Nagy (2020)

Commentationes Mathematicae Universitatis Carolinae

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An algebraic structure is said to be congruence permutable if its arbitrary congruences $α$ and $β$ satisfy the equation $α \circ β = β \circ α$ , where $\circ$ denotes the usual composition of binary relations. To an arbitrary $G$ -set $X$ satisfying $G \cap X = \emptyset$ , we assign a semigroup $(G, X, 0)$ on the base set $G \cup X \cup {0}$ containing a zero element $0 \notin G \cup X$ , and examine the connection between the congruence permutability of the $G$ -set $X$ and the semigroup $(G, X, 0)$ .

Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong (2003)

Colloquium Mathematicae

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Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $(f (x_{i}, x_{i}))$ having f evaluated at the greatest common divisor $(x_{i}, x_{i})$ of...

On the Separation Dimension of $K_{ω}$

Yasunao Hattori, Jan van Mill (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that $t r t K_{ω} > ω + 1$ , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

A class number criterion for the equation $(x^{p} - 1) / (x - 1) = p y^{q}$

Benjamin Dupuy (2007)

Acta Arithmetica

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Displaying similar documents to “On the existence of dimension zero divisors in algebraic function fields defined over q ”

Displaying similar documents to “On the existence of dimension zero divisors in algebraic function fields defined over $_{q}$ ”