On the Waring problem with multivariate Dickson polynomials (Q2869135)

Write NEWLINE\[NEWLINED^{(1)}_{e}(x_{1},\cdots,x_{k},a)=u^{e}_{1}+\cdots+u^{e}_{k+1},\, x_{i}=s_{i}(u_{1},\cdots,u_{k+1}),NEWLINE\]NEWLINE where \(s_{i}(u_{1},\cdots,u_{k+1})\) is the \(i\)-th symmetric function in the indeterminates \(u_{1},\cdots,u_{k+1},\) and \(u_{1}\cdots u_{k+1}=a\).NEWLINENEWLINEIn this paper, the authors consider the question of the existence and estimation of the smallest positive integer \(g=g_{a}(e,k,q)\) such that the equation NEWLINE\[NEWLINED^{(1)}_{e}(x_{1,1},\cdots,x_{1,k},a)+\cdots+D^{(1)}_{e}(x_{g,1},\cdots,x_{g,k},a)=c, \, x_{i,j}\in {\mathbb F}_{q},NEWLINE\]NEWLINE is solvable for any \(c\in {\mathbb F}_{q}\).NEWLINENEWLINEWe call \(g_{a}(e,k,q)\) the Waring number of \(D^{(1)}_{e}\). The following result is obtained:NEWLINENEWLINETheorem. Let \(1\leq k_{0}\leq k\) be minimal such that NEWLINE\[NEWLINEgcd(e,(q^{k_{0}+1}-1)/(q-1))\leq\frac{3}{8\sqrt{2}}q^{1/2}.NEWLINE\]NEWLINE If \(a\) is a \((k+1)\)-th power in \({\mathbb F}^{\times}_{q}\), then NEWLINE\[NEWLINEg_{a}(e,k,q)\leq \lceil\frac{8(k_{0}+1)}{\lfloor(k+1)/(k_{0}+1)\rfloor}\rceilNEWLINE\]NEWLINE and otherwise if \(k>k_{0}+1\), NEWLINE\[NEWLINEg_{a}(e,k,q)\leq \lceil\frac{8(k_{0}+1)}{\lfloor k/(k_{0}+1)\rfloor}\rceil .NEWLINE\]NEWLINENEWLINENEWLINEFor the entire collection see [Zbl 1253.00023].