Relative Closure and the Complexity of Pfaffian Elimination

We introduce the “relative closure” operation on one-parametric families ofsemi-Pfaffian sets. We show that finite unions of sets obtained with this operation(“limit sets”) constitute a structure,i.e., a Boolean algebra closed under projections. Any Pfaffian expression, i.e., an expression with Boolean operations, quantifiers, equations and inequalities between Pfaffian functions, defines a limit set. The structure of limit sets is effectively o-minimal: there is an upper bound on the complexity of a limit set defined by a Pfaffian expression,in terms of the complexities of the expression and the Pfaffian functions in it.