# Arc Spaces and Additive Invariants in Real Algebraic and Analytic Geometry

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*Michel Coste; Toshizumi Fukui; Krzysztof Kurdyka; Clint McCrory; Adam Parusiński; Laurentiu Paunescu*

A publication of the Société Mathématique de France

In this volume the authors present some new trends in real
algebraic geometry based on the study of arc spaces and additive
invariants of real algebraic sets. Generally, real algebraic geometry
uses methods of its own that usually differ sharply from the more
widely known methods of complex algebraic geometry. This feature is
particularly apparent when studying the basic topological and
geometric properties of real algebraic sets; the rich algebraic
structures are usually hidden and cannot be recovered from the
topology. The use of arc spaces and additive invariants partially
obviates this disadvantage. Moreover, these methods are often
parallel to the basic approaches of complex algebraic geometry.

The authors' presentation contains the construction of local
topological invariants of real algebraic sets by means of
algebraically constructible functions. This technique is extended to
the wider family of arc-symmetric semialgebraic sets. Moreover, the
latter family defines a natural topology that fills a gap between the
Zariski topology and the euclidean topology.

In real equisingularity theory, Kuo's blow-analytic equivalence of
real analytic function germs provides an equivalence relation that
corresponds to topological equivalence in the complex analytic
set-up. Among other applications, arc-symmetric geometry, via the
motivic integration approach, gives new invariants of this
equivalence, allowing some initial classification results.

The volume contains two courses and two survey articles that are designed
for a wide audience, in particular students and young researchers.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.