On systems of linear and diagonal equation of degree \(p^i+1\) over finite fields of characteristic \(p\)
The object of the paper is to compute an analogue of the ``g'' of the waring problem over finite fields in a special case. More precisely, given positive integers \(k,d\) and a power \(q\) of a prime \(p.\) Let \(G = g(k,d,q)\) be the smallest positive integer such that given two elements \(a,b\) of \(\mathbb{F}_q\) one of them is a sum of \(G\) \(k\)-th powers while the other is a sum of \(G\) \(d\)-th powers. In the special case when \(k=1\) and \(d-1\) is a \(p\)-th power they got the value of \(G.\) Namely, \(p\) odd implies \(G=3\) iff \(q \neq (d-1)^2.\) The study of the particular polynomial \(P =(ax+b)x^q+(bx+d)\) is key. Should be interesting to settle the case \(p=2.\) Moreover, we may wonder what happens when \(k>1\).
scientific article; zbMATH DE number 5316919
scientific article