Opened 3 years ago
Last modified 18 months ago
#19865 new enhancement
Added functionality to compute zeta functions of nondegenerate hypersurfaces over finite fields
Reported by:  malcolmkotok  Owned by:  

Priority:  major  Milestone:  sage7.0 
Component:  number theory  Keywords:  zeta, padic, Lfunction, sd87, sd91 
Cc:  malcolmkotok, ursula  Merged in:  
Authors:  Heidi Goodson, Malcolm Kotok, Renate Scheidler, Mckenzie West, Ursula Whitcher  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  u/mwest/ticket_19865 (Commits)  Commit:  c6c8c8ec15cb66e6dd7833e4366450cc7777bb88 
Dependencies:  Stopgaps: 
Description (last modified by )
As part of his Ph.D. thesis, Malcolm Kotok implemented a function to compute zeta functions of nondegenerate hypersurfaces over finite fields, based on a paper of Sperber and Voight. We wish to incorporate this into Sage. For more information see: http://arxiv.org/abs/1112.4881
Change History (13)
comment:1 Changed 3 years ago by
 Cc malcolmkotok added
comment:2 Changed 3 years ago by
comment:3 Changed 3 years ago by
Note also ticket #15239 regarding testing for nondegeneracy, which might be worth dealing with first.
comment:4 Changed 21 months ago by
 Keywords sd87 added
comment:5 Changed 19 months ago by
For (my own) convenience, the link to the code is http://hdl.handle.net/1802/30832.
comment:6 Changed 19 months ago by
 Keywords sd91 added
comment:7 Changed 19 months ago by
 Cc ursula added
 Description modified (diff)
comment:8 Changed 19 months ago by
I tried this code about a year ago and it seemed to have some serious problems like it not finishing for a plane quartic curve over F_p (for any p). I wrote to Kotok about this, but never heard back from him. It could be some trivial issue, but it is hard to say.
comment:9 Changed 18 months ago by
My understanding is that some successful experiments with the code were made at SD91. Maybe someone can push the result to trac so that the rest of us can help with stresstesting?
comment:10 Changed 18 months ago by
Ok, here is what I'm talking about:
load("ffzeta.sage"); R.<x,y>=PolynomialRing(ZZ,2); # does work: Q=y^2(x^3+x+1); p=13; ffzeta(Q,p,verbose=True); # does not work: Q=y^4+(5*x4)*y^3+(3*x^2+2*x3)*y^2+(x^3+4*x^22*x+3)*y+(3*x^42*x^34*x^2+2*x+2); p=11; ffzeta(Q,p,verbose=True);
comment:11 Changed 18 months ago by
The SperberVoight algorithm being implemented here has as a complexity parameter the number of interior monomials, so it probably doesn't stand much of a chance for dense equations (although descending it from Sage to Cython might help near the borderline).
For smooth projective hypersurfaces, we also have tickets #20265 (deformation) and #23863 (controlled reduction). For nondegenerate toric hypersurfaces, Edgar Costa has functioning C code doing controlled reduction, but I believe it is not yet available for public consumption.
comment:12 Changed 18 months ago by
 Branch set to u/mwest/ticket_19865
comment:13 Changed 18 months ago by
 Commit set to c6c8c8ec15cb66e6dd7833e4366450cc7777bb88
We have some debugging to make this a method for polynomials. There are definitely errors in the computation in the multivariate example at the beginning of the function.
New commits:
30f7aef  New function zeta_function has been added to multi_polynomial

c6c8c8e  Kotok code added, zeta_function is a method of polynomials

There is also a project by Costa, Harvey, and myself to do a different computation of zeta functions of nondegenerate toric hypersurfaces, using MonskyWashnitzer cohomology in place of Dwork's series method. It's not yet clear how these two methods will compare (the SperberVoight method is optimized towards sparse hypersurfaces somewhat more than our approach), so it would be valuable to have both!