Elimination of Constants from Machines over Algebraically Closed Fields

Let K be an algebraically closed field of characteristic 0. We show that constants can be removed efficiently from any machine over K solving a problem which is definable without constants. This gives a new proof of the transfer theorem of Blum, Cucker, Shub & Smale for the problem P ? = NP. We have similar results in positive characteristic for non-uniform complexity classes. We also construct explicit and correct test sequences (in the sense of Heintz and Schnorr) for the class of polynomials which are easy to compute. An earlier version of this paper appeared as NeuroCOLT Technical Report 96-43. The present paper contains in particular a new bound for the size of explicit correct test sequences. 1 A part of this work was done when the author was visiting DIMACS at Rutgers University. 1 Introduction As in discrete complexity theory, the problem P ? = NP is a major open problem in the Blum-Shub-Smale model of computation over the reals [3]. It has been possible to show that P..