We consider dominant, generically algebraic (e.g. generically finite), and tamely ramified (if the characteristic is positive) morphisms of -schemes, where are Nœtherian and integral and is a Krull scheme (e.g. normal Nœtherian), and study the sheaf of tangent vector fields on that lift to tangent vector fields on . We give an easily computable description of these vector fields using valuations along the critical locus. We apply this to answer the question when the liftable derivations can be defined by a tangency condition along the discriminant. In particular, if is a blow-up of a coherent ideal , we show that tangent vector fields that preserve the Ratliff-Rush ideal (equals for high ) associated to are liftable, and that all liftable tangent vector fields preserve the integral closure of . We also generalise in positive characteristic Seidenberg's theorem that all tangent vector fields can be lifted to the normalisation, assuming tame ramification.

Given a set in a Banach space , we define: the tangent set, and the quasi-tangent set to at , concepts more general than the one of tangent vector introduced by Bouligand (1930) and Severi (1931). Both notions prove very suitable in the study of viability problems referring to differential inclusions. Namely, we establish several new necessary, and even necessary and sufficient conditions for viability referring to both differential inclusions and semilinear evolution inclusions, conditions expressed in terms of the tangency concepts introduced.