Montes Algorithm In Function Fields

Abstract:

Let A=k[t]A=k[t] be the polynomial ring over the perfect field kk and f∈A[x]f∈A[x] be a monic irreducible separable polynomial. Denote by F/kF/k the function field determined by ff and consider a given non-zero prime ideal pp of AA. The Montes algorithm determines a new representation, so called OM-representation, of the prime ideals of the (finite) maximal order of FF lying over pp. This yields a new representation of places of function fields.  In this talk we summarize briefly some applications of this new representation; that are the computation of the genus, the computation of the maximal order, and the improvement of the computation of Riemann-Roch spaces.