Places on algebraic curves

Give L an algebraic extension of k(x) where x is an indeterminate, we have defined a lattice Val(L, k) which is a point-free description of the Riemann surface X associated to L/k. (The topology is such that the open correspond to the cofinite sets. Furthermore there is an extra point which corresponds to the trivial valuation ring L.) We present here a fundamental algorithm which of L enumerates all places P where a given non zero element f of L satisfies vP (f) > 0 (that is, all places P where f is zero). From this algorithm follows for instance that the lattice Val(L, k) is decidable. It shows also how to represent any divisor of X as a formal sum of places. (Surprisingly it does not seem possible to associate a place to an arbitrary point of X.) We can also use this algorithm to define what are the poles of a given differential over X. To simplify we suppose that k is algebraically closed. Using the technique of [2] we know how to make constructive sense of this assumption. In practice it means that when we have a polynomial we introduce new symbols with constraints that they have to be a root of this polynomial. These symbols are treated uniformely until some questions about them (for instance are they also root of another polynomial?) partition them in smaller groups. In [5, 6], Edwards does not assume the field of constants to be algebraically closed but introduces instead extension when needed. It seems simpler to work from an algebraically closed field and to interpret the computations over it in a dynamic way. Since all computations are done in term of polynomials (without having to decide irreducibility but only computing gcd) it seems likely that the main results (for instance decidability of the lattice Val(L, k)) hold without the hypothesis that k is algebraically closed. We assume that L is determined by an equation χ(x, y) = yn + p1(x)y + . . .+ pn(x) = 0 where χ(X,Y ) = Y n + p1(X)Y n−1 + . . . + pn(X) is a polynomial of k[X,Y ] irreducible in k(X)[Y ].