semianalytic

Semianalytic is a term used in real analytic geometry to describe objects, typically sets or functions, that can be locally described using real analytic data. A subset X of a real analytic manifold is semianalytic if, for every point p in X, there exists a neighborhood U of p such that X ∩ U is a finite union of sets of the form {x in U | f1(x) = 0, ..., fk(x) = 0, g1(x) > 0, ..., gl(x) > 0}, where the fi and gj are real analytic on U. Put differently, semianalytic sets are constructed from analytic equalities and strict inequalities through finite unions and intersections.

Semianalytic functions are those whose graphs are semianalytic subsets of the product space, or more broadly, functions that are analytic on each piece of a semianalytic partition of their domain. This class contains real analytic functions and, in general, is larger than the class of semialgebraic functions, which are described using only polynomial equations and inequalities.

Related concepts include subanalytic sets, which are projections of semianalytic sets and often arise in more general geometric contexts, and o-minimal structures, which provide a framework for taming the complexity of definable sets in real analytic settings.

The notion is used to study local and global geometric properties of analytic varieties, stratifications, resolution of singularities, and certain model-theoretic questions in real analytic logic. Precise definitions and conventions can vary among authors, but the core idea remains: semianalytic objects are locally described by finitely many analytic equations and inequalities.