On invariance of degree for certain computations

We prove that a sequential or parallel machine in the Blum-Shub-Smale model, which recognizes the roots of an irreducible polynomial f (x, y) of degree n, in globally bounded time T, must actually compute a polynomial of degree n. This generalizes results of Cucker, and holds for polynomials and machines over an infinite ordered topological field and in certain cases over Z as well. It follows that in the Blum-Shub-Smate model with unit height, P not equal NC for machines over Z. (C) 2004 Elsevier Inc. All rights reserved.