WHEN IS A DISTRIBUTION OF SIGNS LOCALLY COMPLETABLE

Let V be an irreducible nonsingular algebraic surface, Y subset-of V b e an algebraic curve and P a point of Y. Suppose a sign distribution i s given locally in a neighbourhood of P on some connected components o f V - Y. We give an algorithmic criterion to decide whether this sign distribution is induced by a regular function or not. As an applicatio n, this criterion enables one to decide whether two semialgebraic sets can be locally separated or not.