Timeline for Computationally bounding a curve's genus from below?
Current License: CC BY-SA 2.5
16 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Apr 3, 2010 at 7:08 | vote | accept | Dror Speiser | ||
| Mar 31, 2010 at 16:57 | comment | added | damiano | Just a silly remark: if the curve has singular points, then all its reductions will also have singular points. As for the possible suggestion you offered: if there are singular points, there are lower order corrections to the Weil bounds and they can have either sign. For instance, if a pair of conjugate points is identified in a node, it gets counted as a "point", if a pair of points defined over the base-field gets identified in a node, then your count is too small. Ultimately, everything boils down to what you can say about the singular points, since they affect the geometric genus! | |
| Mar 31, 2010 at 16:32 | comment | added | Dror Speiser | @Buzzard: I did the exact same thing today! But, I also let magma calculate the genus over characteristic 0 - which did not end, same as in Singular. I am guessing both programs use Groebner bases, which take a long time to calculate. The problem with reducing modulo is that your chosen prime might be bad, and the method becomes heuristic. But as Felipe said, this doesn't increase genus. For the current question, this is the best answer. First one to post it gets my vote. | |
| Mar 31, 2010 at 11:58 | answer | added | singe | timeline score: 0 | |
| Mar 31, 2010 at 9:37 | answer | added | damiano | timeline score: 3 | |
| Mar 31, 2010 at 8:47 | comment | added | Kevin Buzzard | @Dror: finding singular points of an equation involves solving three equations in two unknowns. This is a grotty Groebner basis calculation, if you like: does Singular have this implemented? I just tried this in magma with a random polynomial of degree 25 in x and 23 in y, modulo a random 3-digit prime, and magma checked that there were no singular points in well under 1 second. | |
| Mar 31, 2010 at 1:24 | answer | added | AVS | timeline score: 2 | |
| Mar 30, 2010 at 23:07 | comment | added | Dror Speiser | I'd have to compute it again, which took a while since sage isn't the best with rings of rings of polynomials... Anyway, this is not about my specific polynomial. The question is general - the fact that Singular couldn't hack it, but you propose you can, in my opinion, should force you to write a new implementation for sage. | |
| Mar 30, 2010 at 23:01 | comment | added | Felipe Voloch | If the coefficients are big, you can work mod p. The genus doesn't increase under reduction mod p. If you want to post the polynomial, I can have a go at it. | |
| Mar 30, 2010 at 22:58 | comment | added | Dror Speiser | Before I thought of factoring $Res_y (f,df/dx)$, but actually it would be much faster to compute $GCD(Res_y(f,df/dx), Res_y(f,df/dy))$. But this isn't trivial either since if the degrees are large, then the coefficients get very big, making computation slow. | |
| Mar 30, 2010 at 22:13 | comment | added | Dror Speiser | @Buzzard: I believe you have. Finding singular points involves, with the simplest algorithm I can think of, factoring a polynomial of degree (d-1)(d-2)/2. In my specific case this is almost 300, which is quite large. | |
| Mar 30, 2010 at 21:52 | comment | added | Kevin Buzzard | This is just an affine plane curve, right? Did you factorize the equation? Did you look for singular points (including at infinity)? Both are very computationally easy. If it factors then you can compute the genus of the pieces. If it's irreducible and smooth then it has genus (d-1)(d-2)/2. If it has a singular point you can blow it up: move it to the origin and draw lines through it of the form y=tx; this decreases the degree, and then you keep doing. Have I missed an issue here? | |
| Mar 30, 2010 at 21:34 | comment | added | Dror Speiser | @Felipe: Since that is equivalent to the question, I'm hoping you or someone else knows how to do that efficiently :) | |
| Mar 30, 2010 at 21:29 | comment | added | Felipe Voloch | Counting points on singular curves can be tricky, the Weil bound applies to a non-singular model. Can you bound the number of singular points? | |
| Mar 30, 2010 at 21:11 | history | asked | Dror Speiser | CC BY-SA 2.5 |