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

Barone, 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 -