305
Comment:

← Revision 3 as of 20081114 13:42:08 ⇥
1574
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 BrillNoether algorithm for plane curves over finite field. See the documentation for examples where singular returns invalid results 
Line 5:  Line 5: 
Integral closure  * BrillNoether is a welldocumented algorithm. There are already master theses around on implementing it. For implementing BrillNoether, one needs to analyze the singularities of the curve, which one can do via blowup. 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 BrillNoether 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 BrillNoether and of Hess's method for comparison. I don't think they have ever properly been pitched against each other. 
Line 13:  Line 13: 
Divisors RiemannRoch spaces Linear 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 BrillNoether algorithm for plane curves over finite field. See the documentation for examples where singular returns invalid results
 BrillNoether is a welldocumented algorithm. There are already master theses around on implementing it. For implementing BrillNoether, one needs to analyze the singularities of the curve, which one can do via blowup. 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 BrillNoether 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 BrillNoether 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.