Elimination of parameters in the polynomial hierarchy

Blum, Cucker, Shub and Smale have shown that the problem "P =NP?" has the same answer in all algebraically closed fields of characteristic 0. We generalize this result to the polynomial hierarchy: if it collapses over an algebraically closed field of characteristic 0, then it must collapse at the same level over all algebraically closed fields of characteristic 0. The main ingredient of their proof was a theorem on the elimination of parameters, which we also extend to the polynomial hierarchy. Similar but somewhat weaker results hold in positive characteristic. The present paper updates a technical report (LIP Research Report 97-37) with the same title, and in particular includes new results on interactive protocols and boolean parts. (C) 1999-Elsevier Science B.V. All rights reserved.