Integral closures and weight functions over finite fields

Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in N^r and the corresponding weighted total-degree monomial orderings lift naturally from one domain R_j-1in the tower to the next, R_j, the integral closure of R_j-1[x_j]/<0(x_j)>. The q-th power algorithm is reworked in this more general setting to produce this integral closure over finite fields, though the application is primarily that of calculating the normalizations of curves related to one-point AG codes arising from towers of function fields. Every attempt has been made to couch all the theory in terms of multivariate polynomial rings and ideals instead of the terminology from algebraic geometry or function field theory, and to avoid the use of any type of series expansion.