A weighted module view of integral closures of affine domains of type I
Douglas A. Leonard  Department of Mathematics and Statistics, Auburn University, Auburn, AL 368495307, United States (email) Abstract: A type I presentation $S=R/J$ of an affine (order) domain has a weight function injective on the monomials in the footprint $\Delta(J)$. This can be extended naturally to a presentation, $\overline{R}/\overline{J}$, of the integral closure $ic(S)$. This presentation over $P$:$=F[x_n,\ldots,x_1]$ as an affine $P$algebra relative to a corresponding grevlexoverweight monomial ordering is shown to have a minimal, reduced GrÃ¶bner basis (for the ideal of relations $\overline{J}$) 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 onepoint AG codes, not only from curves (with $P=F[x_1]$) but also from higherdimensional varieties.
Keywords: AG codes, integral closure, normalization, order domains.
Received: June 2008; Revised: January 2009; Available Online: January 2009. 
2016 Impact Factor.8
