Equations in the Hadamard ring of rational functions

Andrea Ferretti; Umberto Zannier

Displaying similar documents to “Equations in the Hadamard ring of rational functions”

Augmentation quotients for Burnside rings of generalized dihedral groups

Shan Chang (2016)

Czechoslovak Mathematical Journal

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Let $H$ be a finite abelian group of odd order, $𝒟$ be its generalized dihedral group, i.e., the semidirect product of $C_{2}$ acting on $H$ by inverting elements, where $C_{2}$ is the cyclic group of order two. Let $Ω (𝒟)$ be the Burnside ring of $𝒟$ , $Δ (𝒟)$ be the augmentation ideal of $Ω (𝒟)$ . Denote by $Δ^{n} (𝒟)$ and $Q_{n} (𝒟)$ the $n$ th power of $Δ (𝒟)$ and the $n$ th consecutive quotient group $Δ^{n} (𝒟) / Δ^{n + 1} (𝒟)$ , respectively. This paper provides an explicit $ℤ$ -basis for $Δ^{n} (𝒟)$ and determines the isomorphism class of $Q_{n} (𝒟)$ for each positive integer $n$ .

The clean elements of the ring $ℛ (L)$

Ali Akbar Estaji, Maryam Taha (2024)

Czechoslovak Mathematical Journal

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We characterize clean elements of $ℛ (L)$ and show that $α \in ℛ (L)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that $𝔠_{L} (coz (α - 1)) \subseteq U \subseteq 𝔬_{L} (coz (α))$ . Also, we prove that $ℛ (L)$ is clean if and only if $ℛ (L)$ has a clean prime ideal. Then, according to the results about $ℛ (L),$ we immediately get results about $𝒞_{c} (L) .$

Invariants, torsion indices and oriented cohomology of complete flags

Baptiste Calmès, Viktor Petrov, Kirill Zainoulline (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $G$ be a split semisimple linear algebraic group over a field and let $T$ be a split maximal torus of $G$ . Let $𝗁$ be an oriented cohomology (algebraic cobordism, connective $K$ -theory, Chow groups, Grothendieck’s $K_{0}$ , etc.) with formal group law $F$ . We construct a ring from $F$ and the characters of $T$ , that we call a formal group ring, and we define a characteristic ring morphism $c$ from this formal group ring to $𝗁 (G / B)$ where $G / B$ is the variety of Borel subgroups of $G$ . Our main result says that when the...

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

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Let $ϕ$ be a substitution of Pisot type on the alphabet $𝒜 = {1, 2, ..., d}$ ; $ϕ$ satisfies theif for every $i, j \in 𝒜$ , there are integers $k, n$ such that $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same $k$ -th letter, and the prefixes of length $k - 1$ of $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d = 2$ and provide a partial result for $d \geq 2$ .

Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_{r}$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n \geq 1$ a fixed positive integer. Let $α$ be an automorphism of the ring $R$ . An additive map $D : R \to R$ is called an $α$ -derivation (or a skew derivation) on $R$ if $D (x y) = D (x) y + α (x) D (y)$ for all $x, y \in R$ . An additive mapping $F : R \to R$ is called a generalized $α$ -derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F (x y) = F (x) y + α (x) D (y)$ for all $x, y \in R$ . We prove...

Symmetric and reversible properties of bi-amalgamated rings

Antonysamy Aruldoss, Chelliah Selvaraj (2024)

Czechoslovak Mathematical Journal

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Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $K$ and $K^{'}$ be two ideals of $B$ and $C$ , respectively, such that $f^{- 1} (K) = g^{- 1} (K^{'})$ . We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A ⋈^{f, g} (K, K^{'})$ of $A$ with $(B, C)$ along $(K, K^{'})$ with respect to $(f, g)$ .

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. ...

Models of group schemes of roots of unity

A. Mézard, M. Romagny, D. Tossici (2013)

Annales de l’institut Fourier

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Let $𝒪_{K}$ be a discrete valuation ring of mixed characteristics $(0, p)$ , with residue field $k$ . Using work of Sekiguchi and Suwa, we construct some finite flat $𝒪_{K}$ -models of the group scheme $μ_{p^{n}, K}$ of $p^{n}$ -th roots of unity, which we call . We carefully set out the general framework and algebraic properties of this construction. When $k$ is perfect and $𝒪_{K}$ is a complete totally ramified extension of the ring of Witt vectors $W (k)$ , we provide a parallel study of the Breuil-Kisin modules of finite flat models of $μ_{p^{n}, K}$ ,...

Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from 2, $Q_{r}$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$ , $G$ are generalized skew derivations of $R$ with the same associated automorphism $α$ , and $p (x_{1}, ..., x_{n})$ is a non-central polynomial over $C$ such that $[F (x), α (y)] = G ([x, y])$ for all $x, y \in {p (r_{1}, ..., r_{n}) : r_{1}, ..., r_{n} \in R}$ . Then there exists $λ \in C$ such that $F (x) = G (x) = λ α (x)$ for all $x \in R$ .

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $C$ be a hyperelliptic curve of genus $g \geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$ . We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$ , $S$ and $K$ . In particular, we obtain that for any given number field $K$ , finite set of places $S$ of $K$ and integer $g \geq 1$ one can in principle determine the set of $K$ -isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

Semicommutativity of the rings relative to prime radical

Handan Kose, Burcu Ungor (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$ -semicommutative. We prove that a ring $R$ is $P$ -semicommutative if and only if $R [x]$ is $P$ -semicommutative if and only if $R [x, x^{- 1}]$ is $P$ -semicommutative. Also, if $R [[x]]$ is $P$ -semicommutative, then $R$ is $P$ -semicommutative. The converse holds provided that $P (R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$ , $R$ is $P$ -semicommutative...