Counting Points of Multivariate Polynomials Over Finite Fields

[Caleb Robelle]

I am using sage version 7.3 on Linux mint 18. I was wondering if there are algorithms to count points of multivariate polynomials over finite fields implemented in sage that were faster than a simple brute force search. I haven't been able to find any on my own. Any advice would be appreciated. Thanks!

[Kwankyu]

On Tuesday, July 9, 2019 at 1:47:50 AM UTC+2, Caleb Robelle wrote:

I am using sage version 7.3 on Linux mint 18. I was wondering if there are algorithms to count points of multivariate polynomials over finite fields implemented in sage that were faster than a simple brute force search. I haven't been able to find any on my own. Any advice would be appreciated. Thanks!

Define the ideal I generated by the polynomials and the field equations (of the form x^q-x with variable x and q order of the finite field). Then I.variety() computes the points using Groebner basis theory. 

[John Cremona]

As far as I know the answer is "no" except for elliptic and hyperelliptic curves.

John Cremona

On Tue, 9 Jul 2019 at 00:47, Caleb Robelle <caro...@umbc.edu> wrote:

I am using sage version 7.3 on Linux mint 18. I was wondering if there are algorithms to count points of multivariate polynomials over finite fields implemented in sage that were faster than a simple brute force search. I haven't been able to find any on my own. Any advice would be appreciated. Thanks!

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