Elimination of Parameters in the Polynomial Hierarchy Ecole Normale Supérieure De Lyon Elimination of Parameters in the Polynomial Hierarchy Elimination of Parameters in the Polynomial Hierarchy

Blum, Cucker, Shub and Smale have shown that the \P = NP ?" problem has the same answer in all algebraically closed elds of characteristic 0. We generalize this result to the polynomial hierarchy: if it collapses over an algebraically closed eld of characteristic 0, then it must collapse at the same level over all algebraically closed elds 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. Blum, Cucker, Shub et Smale ont montr e que la r eponse au prob-l eme \P = NP ?" est la m^ eme dans tous les corps alg ebriquement clos de caract eristique 0. Nous g en eralisons ce r esultat a la hi erarchie polynomiale: si elle s'eeondre pour un corps alg ebriquement clos de caract eristique 0, alors elle s'eeondre au m^ eme niveau pour tous les corps alg ebriquement clos de caract eristique 0. L'ingr edient principal de leur d emonstration est un th eor eme d' elimination des param etres, que nous etendons egalement a la hi erarchie polynomiale. Des r esul-tats similaires mais un peu plus faibles s'appliquent en caract eristique positive. Abstract Blum, Cucker, Shub and Smale have shown that the \P = NP ?" problem has the same answer in all algebraically closed elds of characteristic 0. We generalize this result to the polynomial hierarchy: if it collapses over an algebraically closed eld of characteristic 0, then it must collapse at the same level over all algebraically closed elds 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.