Abstract:Let F q stand for the finite field of odd characteristic p with q elements (q = p n , n ∈ N) and F * q denote the set of all the nonzero elements of F q . Let m and t be positive integers. In this paper, by using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over F q :where the integers t > 0, r 0 = 0 < r 1 < r 2 < ... < r t , 1 ≤ n 1 < n 2 < ... < n t , 0 ≤ j ≤ t − 1, b k ∈ F q , a k,i ∈ F * q , (… Show more
“…Meanwhile, they proposed an interesting question which was recently answerd by Hu and Hong [15]. A more general question was suggested by Hu, Hong and Zhao in [16] that can be stated as follows.…”
mentioning
confidence: 99%
“…But it is kept open when m ≥ 2. Clearly, Yang [36], Song and Chen [25] and Hu and Hong [15] gave a partial answer to Problem 1.1 when m ≥ 2.…”
mentioning
confidence: 99%
“…). One knows that H l (see [15]) is independent of the choice of the primitive element α. In what follows, we let N stand for the number of rational points on the algebraic variety defined by (1.4).…”
2) j,n 4 n4 − b 2 = 0, where the integers 1 ≤ r 1 < r 2 , 1 ≤ r 3 < r q (1 ≤ i ≤ r 2 ), a 2j ∈ F * q (1 ≤ j ≤ r 4 ) and the exponent of each variable is positive integer. An example is also presented to demonstrate the validity of the main result.
“…Meanwhile, they proposed an interesting question which was recently answerd by Hu and Hong [15]. A more general question was suggested by Hu, Hong and Zhao in [16] that can be stated as follows.…”
mentioning
confidence: 99%
“…But it is kept open when m ≥ 2. Clearly, Yang [36], Song and Chen [25] and Hu and Hong [15] gave a partial answer to Problem 1.1 when m ≥ 2.…”
mentioning
confidence: 99%
“…). One knows that H l (see [15]) is independent of the choice of the primitive element α. In what follows, we let N stand for the number of rational points on the algebraic variety defined by (1.4).…”
2) j,n 4 n4 − b 2 = 0, where the integers 1 ≤ r 1 < r 2 , 1 ≤ r 3 < r q (1 ≤ i ≤ r 2 ), a 2j ∈ F * q (1 ≤ j ≤ r 4 ) and the exponent of each variable is positive integer. An example is also presented to demonstrate the validity of the main result.
Let 𝔽q be the finite field of q elements and f be a nonzero polynomial over 𝔽q. For each b ϵ 𝔽q, let Nq(f = b) denote the number of 𝔽q-rational points on the affine hypersurface f = b. We obtain the formula of Nq(f = b) for a class of hypersurfaces over 𝔽q by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.