Allan writes: << I am guessing that a lot of the important struts supporting the theory of algebraic curves would be severely disrupted by the admission of non-polynomials. Most of the theory pivots on results that are in some sense generalizations of the root-counting corollaries of the Fundamental Theorem of Algebra, and root-counting is the first thing that fails when you leave the comfy confines of polynomials. I doubt if there could be any such thing as an analytic Riemann-Roch theorem.
No doubt many things fall apart when one goes beyond polynomials. I think Chris has been asking about real-valued functions, but for complex analytic functions there is indeed a Riemann-Roch theorem. (See, e.g., Theorem 16.9 in "Lectures on Riemann Surfaces" by Otto Forster, 1981). --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele