A measure for semialgebraic sets related to Boolean complexity (Q1262914)

There are several complexity measures for semialgebraic sets (over a real closed field R). The author considers besides affine semialgebraic sets also ``spherical'' ones, which are defined by homogeneous polynomial inequalities on \(S^ N:=(R^{N+1}-\{0\})/R^+\). He introduces a complexity measure, which is easy to handle in the spherical case, where he can define a complexity reducing operator. He shows how to apply results on spherical geometry to affine geometry and gives a ``lower bound for the number of inequalities necessary to define certain affine semialgebraic sets''.