Zeros of a system of diagonal polynomials over finite fields

Abstract

Let $F_q$ be the finite field of characteristic p, having $q=p^a$ elements and let $F_q^{⁎}$ be the unit group of $F_q$. Let $N\left(V\right)$ be the number of $F_q$-rational points of the affine algebraic variety defined by the simultaneous vanishing of the diagonal polynomials $f_i\left(x\right)=a_{i1}{x}_1^{d_{i1}}+\cdots +a_{in}{x}_n^{d_{in}}+c_i$, $i=1,\dots ,r$, where $a_{ij}\in F_q^{⁎}$, $c_i\in F_q$ and $d_{ij}$ is a nonnegative integer for $1\le i\le r,1\le j\le n$. By using properties of Teichmüller representations and the Stickelberger relation applied by Ax and Wan, we show that${\mathrm{ord}}_{q}N\left(V\right)\ge \lceil \sum _{j=1}^n\frac{1}{\underset{1\le i\le r}{\max }\left\{d_{ij}\right\}}\rceil -r$ if ${\max }_{1\le i\le r}\left\{d_{ij}\right\}>0$ for all integers j with $1\le j\le n$, this improves Cao's result which announces the same statement under the condition $d_{ij}>0$ for all integers $i,j$ with $1\le i\le r,1\le j\le n$. For any nonnegative integer m, let $\sigma \left(m\right)$ be the digital sum of m in base p. Then we set up a p-adic version of the first estimate that${\mathrm{ord}}_{p}N\left(V\right)\ge \lceil a\sum _{j=1}^n\frac{1}{\underset{1\le i\le r}{\max }\left\{\sigma \right(d_{ij}\left)\right\}}\rceil -ar,$ which generalizes Moreno and Castro's result from one diagonal polynomial to a system of diagonal polynomials. This also improves the Ax-Katz-Moreno-Moreno theorem in certain cases. Moreover, we extend the study to a more general variety defined by a system of generalized diagonal polynomials.