I am trying to get a rough overview of the best methods one might use to find solutions of multivariate polynomial equations over large finite fields. We can suppose for simplicity that the given system has a finite number of solutions (over the closure).
As I understand it, the general algebraic methods for finding solutions over rational numbers are either based on the Grobner basis (computing the elimination ideal) or are based on the resultants. First question: Are there any other general algebraic methods?
Second (main) question: Is the situation over finite fields any different? Are there any other general methods which only work for finite fields?
To maybe illustrate my confusion from all the sources I have been going through: The figure on page 3 of this thesis on finite field multivariate polynomial equations is trying to give the reader some rough overview. But it doesn't include any resultant-based methods. Are these not suitable for finite fields for some reason I am not seeing?
Any answers or sources to answers would be very appreciated.