Recent zbMATH articles in MSC 14Phttps://zbmath.org/atom/cc/14P2023-01-20T17:58:23.823708ZWerkzeugOn the Reeb spaces of definable mapshttps://zbmath.org/1500.030142023-01-20T17:58:23.823708Z"Basu, Saugata"https://zbmath.org/authors/?q=ai:basu.saugata"Cox, Nathanael"https://zbmath.org/authors/?q=ai:cox.nathanael"Percival, Sarah"https://zbmath.org/authors/?q=ai:percival.sarahThis paper studies the Reeb space of a continuous map \(f:X \to Y\) definable in an o-minimal expansion of an ordered real closed field. The Reeb space of \(f\) denoted by \(\operatorname{Reeb}(f)\) is the topological space \(X/\sim\) equipped with the quotient topology, where the equivalence relation \(\sim\) is defined so that \(x \sim x'\) if and only if \(f(x)=f(x')\) and both \(x\) and \(x'\) are contained in the same definably connected component of \(f^{-1}(f(x))\).
The first contribution of this paper is the assertion that the Reeb space of \(f\) exists as a definably proper quotient of \(X\) when \(X\) is closed and bounded in its ambient space. Its proof is constructive, but its known complexity is at least doubly exponential. This paper does not provide a singly exponential algorithm for constructing the Reeb space, but alternatively gives the upper bounds of the sum of its Betti numbers \(b(\operatorname{Reeb}(f))\) of the Reeb space. The paper firstly introduces the negative result that \(b(\operatorname{Reeb}(f_n))\) is arbitrarily larger than \(b(X_n)\) for some sequences of maps \(( f _n: X _n \to Y_n)_{n>0}\). Its second result is a positive one. It gives a singly exponential upper bound on the sum of the Betti numbers of the Reeb space of a proper semi-algebraic map in terms of the number and degrees of the polynomials defining the map.
Reviewer: Fujita Masato (Kure)Principal kinematic formulas for germs of closed definable setshttps://zbmath.org/1500.530842023-01-20T17:58:23.823708Z"Dutertre, Nicolas"https://zbmath.org/authors/?q=ai:dutertre.nicolasThe author proves two principal kinematic formulas for germs of closed definable sets in \(\mathbb{R}^n\), that generalize the Cauchy-Crofton formula for the density due to Comte and the infinitesimal linear kinematic formula. The paper is very clear, well written, and quite interesting.
Reviewer: Euripedes Carvalho da Silva (Maracanaú)