Geodesic
On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials
Journal of the European Mathematical Society, Tome 16 (2014) no. 8, pp. 1527-1554.

Voir la notice de l'article provenant de la source EMS Press

We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order formula defining the family. As an important consequence, we derive that the number of distinct homotopy types of semi-algebraic subsets of Rk defined by a quantifier-free first order formula Φ, where the sum of the additive complexities of the polynomials appearing in Φ is at most a, is bounded by 2(k+a)O(1). This proves a conjecture made in [5].
DOI : 10.4171/jems/468
Classification : 14-XX, 00-XX, 68-XX
Keywords: Semi-algebraic sets, additive complexity, homotopy types, Hausdorff limit 
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     title = {On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials},
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Sal Barone; Saugata Basu. On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials. Journal of the European Mathematical Society, Tome 16 (2014) no. 8, pp. 1527-1554. doi : 10.4171/jems/468. http://geodesic.mathdoc.fr/articles/10.4171/jems/468/

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