A weighted module view of integral closures of affine domains of type I

A type I presentation S = R/J of an affine (order) domain has a weight function injective on the monomials in the footprint (J). This can be extended naturally to a presentation, R/J, of the integral closure ic(S). This presentation over P := F(xn, . . . , x1) as an affineP-algebra relative to a corresponding grevlex-over-weight monomial ordering is shown to have a minimal, reduced Grobner basis (for the ideal of relations J) consisiting only of P-quadratic relations defining the multiplication of the P-module generators and possibly some P-linear relations if those generators are not independent over P. There then may be better choices for P to minimize the number of P-module generators needed. The intended coding theory application is to the description of one-point AG codes, not only from curves (with P = F(x1)) but also from higher-dimensional varieties. To properly describe a curve X to be used to define a one-point AG code, it is necessary to put it in special position relative to that one special point P∞, with variables corresponding to rational homogeneous functions modulo X, with no poles except possibly atP∞. Then the generator and/or parity-check functions come from the vector space L(mP∞) of said functions with pole order at most m, contained in the ring L(∞P∞) of all such functions. The footprint (J) of the affine domain S = R/J of type I defining the curve does not usually define all of L(∞P∞), but the footprint of its integral closure (in its field of fractions) does. So there is a compelling reason to study integral closures in the context of AG coding. If one views the pole orders as corresponding to the (negatives of the) trailing exponents in the Laurent series expansions in terms of a local parameter t∞ at P∞, then the obvious generalization to n-dimensional surfaces is in terms of the trailing exponent vector ∈ Nn0 in an expansion involving n independent local parameters.