Waring’s problem for polynomial rings and the digit sum of exponents

摘要

Abstract Let $$overline{mathbb {F}}_p$$ F ˉ p be the algebraic closure of the finite field $$mathbb {F}_p$$ F p . In this paper we develop methods to represent arbitrary elements of $$overline{mathbb {F}}_p[t]$$ F ˉ p [ t ] as sums of perfect k -th powers for any $$kin mathbb {N}$$ k ∈ N relatively prime to p . Using these methods we establish bounds on the necessary number of k -th powers in terms of the sum of the digits of k in its base- p expansion. As one particular application we prove that for any fixed prime $$p 2$$ p 2 and any $$epsilon 0$$ ? 0 the number of $$(p^r-1)$$ ( p r - 1 ) -th powers required is $$mathcal {O}left( r^{(2+epsilon )ln (p)}ight) $$ O r ( 2 + ? ) ln ( p ) as a function of r .