Complexity Lower Bounds for Approximation Algebraic Computation Trees

We prove lower bounds for approximate computations of piecewise polynomial functions which, in particular, apply for round-oo computations of such functions. The goal of this paper is to prove lower bounds for approximated computations. As it is customary for lower bounds, we consider some form of algebraic tree as our computational model (cf. B urgisser, Clausen, and Shokrollahi 1996] or Blum, Cucker, Shub, and Smale 1998] for algebraic trees). But, unlike the usual proofs of lower bounds, which deal with decision problems, we will consider computations of real functions. That is, we consider trees computing functions f : IR n ! IR and, also unlike the usual results on lower bounds, we will allow for approximate computations. To understand the nature of our results let us look rst at an example.