Generalized positivstellensatz for valued fields

We give a generalized form of the Positivstellensatz for algebraically clos ed valued fields: there is an algebraic certificate when a "basic semialgeb raic subset" of L-n is empty (where L is an agebraically closed valued fiel d). Here a basic semialgebraic subset of L-n is given by a finite system of "valued sign conditions" on polynomials: being of valuation > 0, less than or equal to 0, = 0, < 0, greater than or equal to 0 = infinity or not equa l infinity. Moreover we give a formal abstract version of the same theorem. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevie r SAS.