Output details
11 - Computer Science and Informatics
University of Bath
Approximation of definable sets by compact families, and upper bounds on homotopy and homology
<13>We establish new singly-exponential upper bounds on the topology (more precisely, Betti-numbers) of semi-algebraic and sub analytic sets, and essentially closes the problem of upper bounds. The importance of this topic to mathematics is explained in [Arnold, Russ. Math. Surv. 2002, 57, 833.] The new technique used in our proof has independent interest, for example in establishing real analogue of Toda's theorem [Basu, Zell, Found. Comput. Math., 10, 2010]. The weaker results, obtained previously by authors, are widely known and used: see [Basu, Pollack, Roy, Algorithms in Real Algebraic Geometry, 2nd ed., Springer, 2006].