Back-and-forth systems for generic curves and a decision algorithm for thelimit theory

It was recently shown that the theories of generic algebraic curves converg e to a limit theory as their degrees go to infinity. In this paper we give quantitative versions of this result and other similar results. In particul ar, we show that generic curves of degree higher than 2(2r) cannot be disti nguished by a first-order formula of quantifier rank r. A decision algorith m for the limit theory then follows easily. We also show that in this theor y all formulas are equivalent to boolean combinations of existential formul as, and give a quantitative version of this result. (C) 2001 Elsevier Scien ce B.V. All rights reserved.