Finding Waring's Number Over Finite Fields

Waring's number is the minimum amount of variables n, needed to guarantee that an equation of the form X₁^d + X₂^d + ... + X_n^d = α, has solution over the natural numbers, for every natural number alpha. We study Waring’s number considering the solutions and α over finite fields. This problem has applications to coding theory. In this case we represent Waring's number as n = δ(d,p,) where p is a prime number that represents the number of elements in the finite field. There are results that say that, given an exponent d, there exist a lower bound p^' such that δ(d,p,) = 2  for all p > p^'. In this research we present improvements for this lower bound for some values of d. The new lower bounds were obtained by designing and implementing a computer algorithm in C++. Our results allow us to complete tables of Waring's numbers for d = 1, ..., 11  and every p.