Arithmetic of Weighted Diagonal Surfaces over Finite Fields

Certain diagonal equations over finite fields

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On Restricted Arithmetic Progressions over Finite Fields

Let A be a subset of Fp , the n-dimensional linear space over the prime field Fp of size at least δN (N = p), and let Sv = P −1(v) be the level set of a homogeneous polynomial map P : Fp → Fp of degree d, for v ∈ Fp . We show, that under appropriate conditions, the set A contains at least cN |S| arithmetic progressions of length l ≤ d with common difference in Sv, where c is a positive constant...

Solving systems of diagonal polynomial equations over finite fields

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the polynomial equations. Our algorithm works in time polynomial in the number of equations and the logarithm of the size of the field, whenever the degree of the pol...

L-functions of Twisted Diagonal Exponential Sums over Finite Fields

Let Fq be the finite field of q elements with characteristic p and Fqm its extension of degree m. Fix a nontrivial additive character Ψ and let χ1, ..., χn be multiplicative characters of Fp. For f(x1, ..., xn) ∈ Fq [x1, x−1 1 , ..., xn, x−1 n ], one can form the twisted exponential sum S∗ m(χ1, ..., χn, f). The corresponding L-function is defined by L(χ1, ..., χn, f ; t) = exp( ∞ ∑ m=0 S∗ m(χ1...

Abelian Surfaces over Finite Fields as Jacobians