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It is known that for determining sets Markov’s property is equivalent to Bernstein’s property. We are interested in finding a generalization of this fact for sets which are not determining. In this paper we give examples of sets which are not determining, but have the Bernstein and generalized Markov properties.
Define for a smooth compact hypersurface of its crumpleness as the ratio , where is the distance from to its central set. (In other words, is the maximal radius of an open non-selfintersecting tube around in We prove that any -dimensional non-singular compact algebraic hypersurface of degree is rigidly isotopic to an algebraic hypersurface of degree and of crumpleness . Here , depend only on , and rigid isotopy means an isotopy passing only through hypersurfaces of degree...
Nel presente lavoro si studiano le applicazioni polinomiali proprie
In particolare si prova:
1) se è un'applicazione polinomiale tale che è compatto per ogni , allora è propria;
2) se è polinomiale a fibra compatta e è chiuso in allora è propria;
3) l'insieme delle applicazioni polinomiali proprie di in è denso, nella topologia , nello spazio delle applicazioni di in .
Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial , where are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x...
On s’intéresse aux difféomorphismes birationnels des surfaces algébriques réelles qui possèdent une dynamique réelle simple et une dynamique complexe riche. On donne un exemple d’une telle transformation sur , mais on montre qu’une telle situation est exceptionnelle et impose des conditions fortes à la fois sur la topologie du lieu réel et sur la dynamique réelle.
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