On the space of morphisms into generic real algebraic varieties

Riccardo Ghiloni

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On the space of real algebraic morphisms

Riccardo Ghiloni (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this Note, we announce several results concerning basic properties of the spaces of morphisms between real algebraic varieties. Our results show a surprising intrinsic rigidity of Real Algebraic Geometry and illustrate the great distance which, in some sense, exists between this geometry and Real Nash one. Let us give an example of this rigidity. An affine real algebraic variety $X$ is rigid if, for each affine irreducible real algebraic variety $Z$ , the set of all nonconstant regular...

A Bogomolov property for curves modulo algebraic subgroups

Philipp Habegger (2009)

Bulletin de la Société Mathématique de France

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Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$ . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

Some remarks about proper real algebraic maps

L. Beretta, A. Tognoli (2000)

Bollettino dell'Unione Matematica Italiana

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Nel presente lavoro si studiano le applicazioni polinomiali proprie $φ : R^{n} \to R^{q} .$ In particolare si prova: 1) se $φ : R^{n} \to R$ è un'applicazione polinomiale tale che $φ^{- 1} (y)$ è compatto per ogni $y \in R$ , allora $φ$ è propria; 2) se $φ : R^{n} \to R^{q}$ è polinomiale a fibra compatta e $φ (R^{n})$ è chiuso in $R^{q}$ allora $φ$ è propria; 3) l'insieme delle applicazioni polinomiali proprie di $R^{n}$ in $R^{q}$ è denso, nella topologia $C^{\infty}$ , nello spazio delle applicazioni $C^{\infty}$ di $R^{n}$ in $R^{q}$ .

On the polynomial-like behaviour of certain algebraic functions

Charles Feffermann, Raghavan Narasimhan (1994)

Annales de l'institut Fourier

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Given integers $D > 0, n > 1, 0 < r < n$ and a constant $C > 0$ , consider the space of $r$ -tuples $\vec{P} = (P_{1} ... P_{r})$ of real polynomials in $n$ variables of degree $\leq D$ , whose coefficients are $\leq C$ in absolute value, and satisfying $\det {(\frac{\partial P_{i}}{\partial x_{i}} (0))}_{1 \leq i, j \leq r} = 1$ . We study the family ${f | V}$ of algebraic functions, where $f$ is a polynomial, and $V = {| x | \leq δ, \vec{P} (x) = 0}, δ > 0$ being a constant depending only on $n, D, C$ . The main result is a quantitative extension theorem for these functions which is uniform in $\vec{P}$ . This is used to prove Bernstein-type inequalities which are again uniform with respect to $\vec{P}$ . ...

Some approximation problems in semi-algebraic geometry

Shmuel Friedland, Małgorzata Stawiska (2015)

Banach Center Publications

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In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set C in the space ℝⁿ endowed with a semi-algebraic norm ν. Under additional assumptions on ν we prove semi-algebraicity of the set of points of unique approximation and other sets associated with the distance to C. For C irreducible algebraic we study the critical point correspondence and introduce the ν-distance degree, generalizing the notion developed by other authors for the Euclidean...

Algebraic approximation of manifolds and spaces

A. Tognoli (1979-1980)

Séminaire Bourbaki

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The mean values of logarithms of algebraic integers

Artūras Dubickas (1998)

Journal de théorie des nombres de Bordeaux

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Let $α$ be an algebraic integer of degree $d$ with conjugates $α_{1} = α, α_{2}, \dots, α_{d}$ . In the paper we give a lower bound for the mean value $M_{p} (α) = \sqrt[p]{\frac{1}{d} \sum_{i = 1}^{d} | log | α_{i} {| |}^{p}}$ when $α$ is not a root of unity and $p > 1$ .

Cycles on algebraic models of smooth manifolds

Wojciech Kucharz (2009)

Journal of the European Mathematical Society

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Every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of $M$ . We study modulo 2 homology classes represented by algebraic subsets of $X$ , as $X$ runs through the class of all algebraic models of $M$ . Our main result concerns the case where $M$ is a spin manifold.

A purely algebraic proof of special cases of Tchebotarev's theorem

J. Wójcik (1975)

Acta Arithmetica

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