Study On Certain Hypersurfaces Of Finite Fields

In this paper, we studied the number of F_q-rational points for several hypersurface over finite filed F_q, and obtained Zeta functions for some of the hypersurface mentioned above. We also discussed the problem of solutions over subfield in finite field. Let F = F_q be a finite field with q = p^f elements, where f ≥ 1, p is an odd prime number.In Chapter One, we discussed certain equations over F.The first type of equation iswhere, d_ij > 0,a_i∈ F^*,b ∈ F, 0 < n₁ < n₂. It's a kind of step equation. The second type of equation iswhere, n is a positive integer, d_ij(1≤ < i,j ≤ n) are non-negative integer, a₁,…… ,a_n ∈ F^* and b ∈ F. Such a kind of equations is called a triangular equation.And the third type of equation is the more general step equations, it's the generalization of the two above kinds of equations.Firstly, we got rational explicit formulas for the number of solutions of those three kinds of equations by applying some combinatorial methods. The result of the first type of equation improves Sun's theorem of Chinese Annals of Mathematics (1997(4)). Furthermore, under the condition of gcd(d₁1……d_nn,q - 1) = 1, the rational explicit formula of triangular equation is very interesting and the idea of solving equation is very innovative. And then we applied some results of character sum for the general triangular equations and get the estimate theorem which is very similar to the classical result of diagonal equation.In Chapter Two, we primary used the results obtained in chapter one to study the Zeta function of the hypersurface corresponding to triangular equation and gave some computable formulas. Finally, we applied those formulas to a given equation.In Chapter Three, we introduced the problem of weak form solutions over subfield. And we gave a non-trivial estimate theorem for number of weak form solutions of the triangular equation on subfields by applying character sum and Weil's estimate of character sum.