# On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 8, page 1527-1554
- ISSN: 1435-9855

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topBarone, Sal, and Basu, Saugata. "On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials." Journal of the European Mathematical Society 016.8 (2014): 1527-1554. <http://eudml.org/doc/277530>.

@article{Barone2014,

abstract = {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 $\mathbb \{R\}^k$ defined by a quantifier-free first order formula $\Phi $, where the sum of the additive complexities of the polynomials appearing in $\Phi $ is at most $a$, is bounded by $2^\{(k+a)^\{O(1)\}\}$. This proves a conjecture made in [5].},

author = {Barone, Sal, Basu, Saugata},

journal = {Journal of the European Mathematical Society},

keywords = {semi-algebraic sets; additive complexity; homotopy types; Hausdorff limit; semi-algebraic sets; additive complexity; homotopy types; Hausdorff limit},

language = {eng},

number = {8},

pages = {1527-1554},

publisher = {European Mathematical Society Publishing House},

title = {On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials},

url = {http://eudml.org/doc/277530},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Barone, Sal

AU - Basu, Saugata

TI - On homotopy types of limits of semi-algebraic sets and additive complexity of polynomials

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 8

SP - 1527

EP - 1554

AB - 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 $\mathbb {R}^k$ defined by a quantifier-free first order formula $\Phi $, where the sum of the additive complexities of the polynomials appearing in $\Phi $ is at most $a$, is bounded by $2^{(k+a)^{O(1)}}$. This proves a conjecture made in [5].

LA - eng

KW - semi-algebraic sets; additive complexity; homotopy types; Hausdorff limit; semi-algebraic sets; additive complexity; homotopy types; Hausdorff limit

UR - http://eudml.org/doc/277530

ER -

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