Cycles on algebraic models of smooth manifolds

Wojciech Kucharz

Displaying similar documents to “Cycles on algebraic models of smooth manifolds”

Algebraic Superconnections, $S U (2 ∣ 1)$ and Electroweak Interactions

R. Coquereaux (1992)

Recherche Coopérative sur Programme n°25

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On the principle of real moduli flexibility: perfect parametrizations

Edoardo Ballico, Riccardo Ghiloni (2014)

Annales Polonici Mathematici

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Let V be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer b (arbitrarily large), there exists a trivial Nash family $= {V_{y}}_{y \in R^{b}}$ of real algebraic manifolds such that V₀ = V, is an algebraic family of real algebraic manifolds over $y \in R^{b} ∖ 0$ (possibly singular over y = 0) and is perfectly parametrized by $R^{b}$ in the sense that $V_{y}$ is birationally nonisomorphic to $V_{z}$ for every $y, z \in R^{b}$ with y ≠ z. A similar result continues to hold if V is a singular real algebraic...

Algebraic independence of the values at algebraic points of a class of functions considered by Mahler

N. Ch. Wass

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This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,..., $θ_{m}$ of complex numbers. Specifically, let K be a number field and let f₁(z),..., $f_{m} (z)$ be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) $f_{j} (z^{b}) = \sum_{i = 1}^{m} f_{i} (z) a_{i j} (z) + b_{j} (z)$ (j = i,...,m)for b ≥ 2, $a_{i j} (z)$ , $b_{j} (z)$ in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the $f_{j} (z$ ) converge at z = α and the $a_{i j} (z)$ , $b_{j} (z)$ are analytic at $z = α, α^{b}, α^{b ²}, . . .$ Then the $θ_{i} = f_{i} (α)$ are algebraically independent...

Diagonalization and rationalization of algebraic Laurent series

Boris Adamczewski, Jason P. Bell (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove a quantitative version of a result of Furstenberg [20] and Deligne [14] stating that the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^{A}$ and height at most $A p^{A}$ , where $A$ is an effective constant that only...

Multiplicatively dependent triples of Tribonacci numbers

Carlos Alexis Ruiz Gómez, Florian Luca (2015)

Acta Arithmetica

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We consider the Tribonacci sequence $T : = {T_{n}}_{n \geq 0}$ given by T₀ = 0, T₁ = T₂ = 1 and $T_{n + 3} = T_{n + 2} + T_{n + 1} + T_{n}$ for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

Polynomial relations amongst algebraic units of low measure

John Garza (2014)

Acta Arithmetica

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For an algebraic number field and a subset $α_{1}, . . ., α_{r} \subseteq_{}$ , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $ℚ [x_{1}, . . ., x_{r}]$ vanishing at the point $(α_{1}, . . ., α_{r})$ .

Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

Hajime Kaneko, Takeshi Kurosawa, Yohei Tachiya, Taka-aki Tanaka (2015)

Acta Arithmetica

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Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $\prod_{\binom{k = 1}{U_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) / (U_{d^{k}}))$ (i=1,...,m) or $\prod_{\binom{k = 1}{V_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) (V_{d^{k}})$ (i=1,...,m) to be algebraically dependent, where $a_{i}$ are non-zero integers and $U_{n}$ and $V_{n}$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_{1}, . . ., a_{m}$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically...

A twisted class number formula and Gross's special units over an imaginary quadratic field

Saad El Boukhari (2023)

Czechoslovak Mathematical Journal

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Let $F / k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_{F}$ be the ring of integers of $F$ and $n \geq 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$ -groups of $O_{F}$ using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of $F$ , which is the cardinal of the finite algebraic $K$ -group $K_{2 n - 2} (O_{F})$ .

On the multiples of a badly approximable vector

Yann Bugeaud (2015)

Acta Arithmetica

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Let d be a positive integer and α a real algebraic number of degree d + 1. Set $α ̲ : = (α, α ², . . ., α^{d})$ . It is well-known that $c (α ̲) : = l i m i n f_{q \to \infty} q^{1 / d} \cdot | | q α ̲ | | > 0$ , where ||·|| denotes the distance to the nearest integer. Furthermore, $c (α ̲) n^{- 1 / d} \leq c (n α ̲) \leq n c (α ̲)$ for any integer n ≥ 1. Our main result asserts that there exists a real number C, depending only on α, such that $c (n α ̲) \leq C n^{- 1 / d}$ for any integer n ≥ 1.

Rational fixed points for linear group actions

Pietro Corvaja (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$ , an affine variety $V$ and a finite map $π : V \to G$ , all defined over a finitely generated field $κ$ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $π (V (κ))$ contains a Zariski dense sub-semigroup $Γ \subset G (κ)$ ; namely, there must exist an unramified covering $p : \tilde{G} \to G$ and a map $θ : \tilde{G} \to V$ such that $π \circ θ = p$ . In the case $κ = ℚ$ , $G = 𝔾_{a}$ is the additive group, we...

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...

Algebraic independence of the generating functions of Stern’s sequence and of its twist

Peter Bundschuh, Keijo Väänänen (2013)

Journal de Théorie des Nombres de Bordeaux

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Very recently, the generating function $A (z)$ of the Stern sequence ${(a_{n})}_{n \geq 0}$ , defined by $a_{0} : = 0, a_{1} : = 1,$ and $a_{2 n} : = a_{n}, a_{2 n + 1} : = a_{n} + a_{n + 1}$ for any integer $n > 0$ , has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A (α)$ for every algebraic $α$ with $0 < | α | < 1$ , and this result was generalized in [6] to the effect that, for the same $α$ ’s, all numbers $A (α), A^{'} (α), A^{''} (α), ...$ are algebraically independent. At about the same time, Bacher...

Algebraic and topological properties of some sets in ℓ₁

Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik (2012)

Colloquium Mathematicae

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For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $\sum_{n = 1}^{\infty} x (n)$ . Guthrie and Nymann proved that E(x) is one of the following types of sets: () a finite union of closed intervals; () homeomorphic to the Cantor set; homeomorphic to the set T of subsums of $\sum_{n = 1}^{\infty} b (n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable....

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.

Relative exactness modulo a polynomial map and algebraic $(ℂ^{p}, +)$ -actions

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

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Let $F = (f_{1}, ..., f_{q})$ be a polynomial dominating map from $ℂ^{n}$ to $ℂ^{q}$ . We study the quotient $𝒯^{1} (F)$ of polynomial 1-forms that are exact along the generic fibres of $F$ , by 1-forms of type $d R + \sum a_{i} d f_{i}$ , where $R, a_{1}, ..., a_{q}$ are polynomials. We prove that $𝒯^{1} (F)$ is always a torsion $ℂ [t_{1}, ..., t_{q}]$ -module. Then we determine under which conditions on $F$ we have $𝒯^{1} (F) = 0$ . As an application, we study the behaviour of a class of algebraic $(ℂ^{p}, +)$ -actions on $ℂ^{n}$ , and determine in particular when these actions are trivial.

Stabilization of monomial maps in higher codimension

Jan-Li Lin, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

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A monomial self-map $f$ on a complex toric variety is said to be $k$ -stable if the action induced on the $2 k$ -cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$ , we can find a toric model with at worst quotient singularities where $f$ is $k$ -stable. If $f$ is replaced by an iterate one can find a $k$ -stable model as soon as the dynamical degrees $λ_{k}$ of $f$ satisfy $λ_{k}^{2} > λ_{k - 1} λ_{k + 1}$ . On the other hand, we give examples of monomial maps $f$ , where...

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. ...