Differences between revisions 1 and 3 (spanning 2 versions)
 ⇤ ← Revision 1 as of 2007-09-30 14:42:25 → Size: 305 Editor: NilsBruin Comment: ← Revision 3 as of 2008-11-14 13:42:08 → ⇥ Size: 1574 Editor: localhost Comment: converted to 1.6 markup Deletions are marked like this. Additions are marked like this. Line 1: Line 1: Figure out what support there is for computing on (plane) curves in software available in Sage Some ideas for implementing divisor arithmetic on curves Line 3: Line 3: See what Singular can do for plane curves * In Singular, there is an implementation of the Brill-Noether algorithm for plane curves over finite field. See the documentation for examples where singular returns invalid results Line 5: Line 5: Integral closure * Brill-Noether is a well-documented algorithm. There are already master theses around on implementing it. For implementing Brill-Noether, one needs to analyze the singularities of the curve, which one can do via blow-up. This is going to be challenging if the singular locus is not split over the base field. Line 7: Line 7: Resolution of singularities * Once analyzing singular places is in place, implementing Brill-Noether might be in interesting exercise. Line 9: Line 9: What base rings are supported? * I expect that the approach of Florian Hess (view the function field of a curve as a finite extension of k(t) rather than as the field of fractions of k[x,y]/( f(x,y) ) will be more efficient in most cases. For that we need integral closure of k[x] in k(x)[y]/( f(x,y)). To deal with the places at infinity, one could either work with the integral closure of k[1/x] in k(x)[y]/ ( f(x,y)) or with a semilocal ring with only the places above (1/x). We will need ideal arithmetic in those rings (both using 2 generator representation and as finite k[t]-modules) Line 11: Line 11: Places * It will be interesting to have reasonable implementations of both Brill-Noether and of Hess's method for comparison. I don't think they have ever properly been pitched against each other. Line 13: Line 13: DivisorsRiemann-Roch spacesLinear equivalence of divisors - Somebody would need to implement finite extensions of arbitrary fields. In particular, univariate polynomial factorization over any field.

Some ideas for implementing divisor arithmetic on curves

• In Singular, there is an implementation of the Brill-Noether algorithm for plane curves over finite field. See the documentation for examples where singular returns invalid results
• Brill-Noether is a well-documented algorithm. There are already master theses around on implementing it. For implementing Brill-Noether, one needs to analyze the singularities of the curve, which one can do via blow-up. This is going to be challenging if the singular locus is not split over the base field.
• Once analyzing singular places is in place, implementing Brill-Noether might be in interesting exercise.
• I expect that the approach of Florian Hess (view the function field of a curve as a finite extension of k(t) rather than as the field of fractions of k[x,y]/( f(x,y) ) will be more efficient in most cases. For that we need integral closure of k[x] in k(x)[y]/( f(x,y)). To deal with the places at infinity, one could either work with the integral closure of k[1/x] in k(x)[y]/ ( f(x,y)) or with a semilocal ring with only the places above (1/x). We will need ideal arithmetic in those rings (both using 2 generator representation and as finite k[t]-modules)
• It will be interesting to have reasonable implementations of both Brill-Noether and of Hess's method for comparison. I don't think they have ever properly been pitched against each other. - Somebody would need to implement finite extensions of arbitrary fields. In particular, univariate polynomial factorization over any field.

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