On Lack of Effectiveness in Semi-algebraic Geometry

It is a fact and a motivation of this meeting, that many interesting constructions in semi-algebraic geometry can be effectively done, that is, roughly speaking, by means of algorithms with computably bounded complexity. It is also a fact that semi-algebraic objects have some effective finiteness properties. Hence the interest in finding reasonably ”fast” algorithms or “sharp” bounds. Neverthless, on the other side of the mainstream, one can be interested in essential lack of effectiveness or finiteness. We are going to see some examples and to discuss shortly some possible basic sources of such a lack.