A Non-Standard Bezout Theorem

This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of projective degree d and degree e curves. The result is particularly interesting in that it holds even when we consider intersections at singular points of curves or when the curves contain non-reduced components. The proof also provides motivation for the fact that tangency is a definable relation for families of curves inside a non-linear 1-dimensional Zariski structure X. This is a crucial ingredient in unpublished work by Peterzil and Zilber that any such Zariski structure interprets a pure algebraically closed field L with X as a definable finite cover.