Algebraic Settings for The

When complexity theory is studied over an arbitrary unordered eld K, the classical theory is recaptured with K = Z2. The fundamental result that the Hilbert Nullstellensatz as a decision problem is NP-complete overK allows us to reformulate and investigate complexity questions within an algebraic framework and to develop transfer principles for complexity theory. Here we show that over algebraically closed elds K of characteristic 0 the fundamental problem \P 6=NP?" has a single answer that depends on the tractability of the Hilbert Nullstellensatz over the complex numbers C . A key component of the proof is the Witness Theorem enabling the elimination of transcendental constants in polynomial time. 1. Statement of Main Theorems We consider the Hilbert Nullstellensatz in the form HN=K: given a nite set of polynomials in n variables over a eld K, decide if there is a common zero over K. At rst the eld is taken as the complex number eld C . Relationships with other elds and with problems in number theory will be developed here. This article is essentially Chapter 6 of our book Complexity and Real Computation (to be published by Springer). Background material can be found in [Blum, Shub, and Smale 1989]. Only machines and algorithms which branch on \h(x) = 0?" are considered here. The symbol is not used. Thus the development is quite algebraic, eventually using properties of the height function of algebraic number theory. A main theme is eliminating constants. The moral is roughly: using transcendental and algebraic numbers doesn't help much in speeding up integer decision problems. Let Q be the algebraic closure of the rational number eld Q. The following will be proved. Theorem 1. If P = NP over C , then P = NP over Q, and the converse is also true. 1991 Mathematics Subject Classi cation. 68Q (Computer Science, Theory of Computation), 11G (Number Theory, Arithmetic Algebraic Geometry). Blum was partially supported by the Letts-Villard Chair at Mills College. Cucker was partially supported by DGICyT PB 920498, the ESPRIT BRA Program of the EC under contracts no. 7141 and 8556, projects ALCOM II and NeuroCOLT. Cucker, Smale, and Shub were partially supported by NSF grants. c 0000 American Mathematical Society 0075-8485/00 $1.00 + $.25 per page