On diagonal equations over finite fields via walks in NEPS of graphs

TL;DR

This paper derives an explicit combinatorial formula for counting solutions to certain diagonal equations over finite fields by analyzing walks in NEPS of graphs, connecting algebraic solutions with graph-theoretic structures.

Contribution

It introduces a novel combinatorial approach linking solutions of diagonal equations over finite fields to walks in NEPS of complete graphs.

Findings

Explicit formula for solutions to diagonal equations over finite fields.

Connection established between algebraic equations and graph walks.

Method applicable to equations with specific parameters.

Abstract

In this paper, we obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=\alpha$ over the finite field $\mathbb{F}_{p^{ab}}$, with $k=\frac{p^{ab}-1}{b(p^a-1)}$ and $b>1$ by using the number of $r$-walks in NEPS of complete graphs.

Equations

$${\alpha }_1{x}_1^{k_1}+\cdots +{\alpha }_s{x}_s^{k_s}=\alpha$$

$$x_1^k+x_2^k+\cdots +x_s^k=\alpha$$

$$\Gamma (k,p^m)=\mathrm{Cay}({\mathbb{F}}_{p^m},R_k)\text{where }{R}_k=\{x^k:x\in {\mathbb{F}}_{p^m}^{\ast }\},$$

$$w_{\Gamma (k,p^m)}(s,x,y)=\frac{1}{k^s}\#\{(x_1,\dots ,x_s)\in ({\mathbb{F}}_{p^m}^{\ast }{)}^s:x_1^k+\cdots +x_s^k=y-x\}.$$

$$x+x_1^k+\cdots +x_s^k=y.$$

$$w_{\Gamma (k,p^m)}(r,x,y)=\frac{1}{k^s}\#\{(x_1,\dots ,x_s)\in ({\mathbb{F}}_{p^m}^{\ast }{)}^s:x+x_1^k+\cdots +x_s^k=y\},$$

$$M=\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{s}{i}\right)N_i\text{if }x\ne y.$$

$$M=1+\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{s}{i}\right)N_i.$$

$$w_G(r,v_i,v_j)=(A(G{)}^r{)}_{i,j}$$

$$A=\sum _{\alpha \in \mathcal{B}}{A}_1^{{\alpha }_1}\otimes \cdots \otimes A_n^{{\alpha }_n},$$

$$w_G(r,v_i,v_j)=\sum _{({\beta }_1,\dots ,{\beta }_r)\in {\mathcal{B}}^r}\prod _{t=1}^n{w}_{G_t}({\beta }_{1t}+\cdots +{\beta }_{rt},{\pi }_t(v_i),{\pi }_t(v_j)),$$

$$(A\otimes B)(C\otimes D)=AC\otimes BD.$$

$$A^{r}=\Big{(}\sum_{\alpha\in\mathcal{B}}A_{1}^{\alpha_{1}}\otimes\cdots\otimes A_{n}^{\alpha_{n}}\Big{)}^{r}=\sum_{(\beta_{1},\ldots,\beta_{r})\in\mathcal{B}^{r}}A_{1}^{\beta_{11}+\cdots+\beta_{r1}}\otimes\cdots\otimes A_{n}^{\beta_{1n}+\cdots+\beta_{rn}},$$

$$(A^r{)}_{i,j}=\sum _{({\beta }_1,\dots ,{\beta }_r)\in {\mathcal{B}}^r}(A_1^{{\beta }_{11}+\cdots +{\beta }_{r1}}\otimes \cdots \otimes A_n^{{\beta }_{1n}+\cdots +{\beta }_{rn}}{)}_{i,j}.$$

$$(A_1^{{\beta }_{11}+\cdots +{\beta }_{r1}}\otimes \cdots \otimes A_n^{{\beta }_{1n}+\cdots +{\beta }_{rn}}{)}_{i,j}=\prod _{t=1}^n(A_t^{{\beta }_{1t}+\cdots +{\beta }_{rt}}{)}_{o_t(i),o_t(j)}$$

$$(A_t^{{\beta }_{1t}+\cdots +{\beta }_{rt}}{)}_{o_t(i),o_t(j)}=w_{G_t}({\beta }_{1t}+\cdots +{\beta }_{rt},{\pi }_t(v_i),{\pi }_t(v_j)).$$

$$w_G(r,v_i,v_j)=\sum _{({\beta }_1,\dots ,{\beta }_r)\in {\mathcal{B}}^r}\prod _{t=1}^n{a}_t(v_i,v_j),$$

$$a_{t}(v_{i},v_{j})=\begin{cases}\frac{m_{t}-1}{m_{t}}\big{(}(m_{t}-1)^{\beta_{1t}+\cdots+\beta_{rt}-1}-(-1)^{\beta_{1t}+\cdots+\beta_{rt}-1}\big{)}&\quad\text{if }\pi_{t}(v_{i})=\pi_{t}(v_{j}),\\[5.69054pt] \frac{1}{m_{t}}\big{(}(m_{t}-1)^{\beta_{1t}+\cdots+\beta_{rt}}-(-1)^{\beta_{1t}+\cdots+\beta_{rt}}\big{)}&\quad\text{if }\pi_{t}(v_{i})\neq\pi_{t}(v_{j}).\end{cases}$$

$$w_{K_{m}}(r,w_{i},w_{j})=\begin{cases}\frac{m-1}{m}\big{(}(m-1)^{r-1}-(-1)^{r-1}\big{)}&\quad\text{if }w_{i}=w_{j},\\[5.69054pt] \frac{1}{m}\big{(}(m-1)^{r}-(-1)^{r}\big{)}&\quad\text{if }w_{i}\neq w_{j}.\end{cases}$$

$$w_{G_1}(r,v,v)=(\frac{1}{2})(2^{r-1}-(-1{)}^{r-1})(3^{r-1}-(-1{)}^{r-1})=\frac{6^{r-1}+(-1{)}^r(2^{r-1}+3^{r-1})+1}{2},$$

$$w_{G_2}(r,v,v)=\sum _{\ell =0}^r\left(\genfrac{}{}{0ex}{}{r}{\ell }\right)(\frac{1}{2})(2^{\ell -1}-(-1{)}^{\ell -1})(3^{r-\ell -1}-(-1{)}^{r-\ell -1})$$

$$w_G(r,v_i,v_j)=\sum _{r_1+\cdots +r_n=r}\frac{r!}{r_1!\cdots r_n!}\prod _{t=1}^n{w}_{G_t}(r_t,{\pi }_t(v_i),{\pi }_t(v_j)),$$

$$w_G(r,v_i,v_j)=\sum _{({\beta }_1,\dots ,{\beta }_r)\in {\mathcal{B}}^r}\prod _{t=1}^n{w}_{G_t}({\beta }_{1t}+\cdots +{\beta }_{rt},{\pi }_t(v_i),{\pi }_t(v_j)),$$

$${\beta }_1+\cdots +{\beta }_r=(r_1,r_2,\dots ,r_n)\in ({\mathbb{Z}}_{\ge 0}{)}^n\text{with}{r}_1+r_2+\cdots +r_n=r.$$

$$w_{H(n,q)}(r,v_i,v_j)=\sum _{r_1+\cdots +r_n=r}\frac{r!}{r_1!\cdots r_n!}\prod _{t=1}^n{a}_t(v_i,v_j),$$

$$a_t(v_i,v_j)=\{\begin{array}{ll}\frac{q-1}{q}((q-1{)}^{r_t-1}-(-1{)}^{r_t-1})& \text{if }{\pi }_t(v_i)={\pi }_t(v_j),\\ \frac{1}{q}((q-1{)}^{r_t}-(-1{)}^{r_t})& \text{if }{\pi }_t(v_i)\ne {\pi }_t(v_j).\end{array}$$

$$x=\sum _{i=0}^{b-1}{c}_i{\omega }^{ik}\mapsto [x]=(c_0,c_1,\dots ,c_{b-1})\in ({\mathbb{F}}_{p^a}{)}^b$$

$$\Gamma (k,p^m)\cong H(b,p^a).$$

$$w_{\Gamma (k,p^m)}(r,x,y)=w_{H(b,p^a)}(r,[x],[y])=\sum _{r_1+\cdots +r_b=r}\frac{r!}{r_1!\cdots r_b!}\prod _{i=1}^b{a}_i(x,y),$$

$$a_i(x,y)=\{\begin{array}{ll}\frac{p^a-1}{p^a}((p^a-1{)}^{r_i-1}-(-1{)}^{r_i-1})& \text{if }[x{]}_i=[y{]}_i,\\ \frac{1}{p^a}((p^a-1{)}^{r_i}-(-1{)}^{r_i})& \text{if }[x{]}_i\ne [y{]}_i.\end{array}$$

Full text

2010 Mathematics Subject Classification. Primary 11G25 ;  Secondary 05C25,05C50, 11T99.

Partially supported by CONICET and SECyT-UNC

1. Introduction

Diagonal equations

A diagonal equation over the finite field $\mathbb{F}_{p^{m}}$ is an equation of the form

[TABLE]

for $\alpha_{i}\in\mathbb{F}_{p^{m}}^{*}$, $\alpha\in\mathbb{F}_{p^{m}}$ for $i=1,\ldots,s$. This kind of equation have been study a lot, the interested reader can see the pioneering work [11], who relates the number of solution in terms of Gauss sums. Other authors have used this Weil’s expression to obtain explicit number of solution for especific $\alpha_{i}$’s and $k_{i}$’s, see [1], [2] [3], [9], [10], [12] [13].

In general, it is difficult to find the explicit number of solution of (1.1), in this work we are going to find an explicit combinatorial solution of (1.1) when $\alpha_{i}=1$ and $k_{i}=k=\frac{p^{ab}-1}{b(p^{a}-1)}$ for all $i$ in the finite fields $\mathbb{F}_{p^{ab}}$ by using a relation between the number of solution of (1.1) and the walks of certain graphs which have a special product structure (NEPS).

NEPS operation

Given a set $\mathcal{B}\subseteq\{0,1\}^{n}$ and graphs $G_{1},\ldots,G_{n}$, the NEPS (non-complete extended psum) of these graphs with respect to the basis $\mathcal{B}$ is the graph $G=\mathrm{NEPS}(G_{1},\ldots,G_{n};\mathcal{B})$, whose vertex set is the cartesian product of the vertex sets of the individual graphs, $V(G)=V(G_{1})\times\cdots\times V(G_{n})$ and two vertices $(x_{1},\ldots,x_{n}),(y_{1},\ldots,y_{n})\in V(G)$ are adjacent in $G$, if and only if there exists some $n$-tuple $(\alpha_{1},\ldots,\alpha_{n})\in\mathcal{B}$ such that $x_{i}=y_{i}$, whenever $\alpha_{i}=0$, or $x_{i},y_{i}$ are distinct and adjacent in $G_{i}$, whenever $\alpha_{i}=1$.

The NEPS operation generalizes a number of known graph products, all of which have in common that the vertex set of the resulting graph is the cartesian product of the input vertex sets. For instance, $\mathrm{NEPS}(G_{1},\ldots,G_{n};\{(1,\ldots,1)\})=G_{1}\otimes\cdots\otimes G_{n}$ is the Kronecker product of the $G_{i}$’s; $\mathrm{NEPS}(G_{1},\ldots,G_{n};\{e_{1},\ldots,e_{n}\})=G_{1}+\cdots+G_{n}$ (where $e_{i}$ is the vector which only $1$ in the position $i$) is the sum of the graphs $G_{i}$; $\mathrm{NEPS}(G_{1},G_{2};\{(1,1),(1,0),(0,1)\})=G_{1}\boxtimes G_{2}$ is the strong product of $G_{1},G_{2}$. We refer to [4] or [5] for the history of the notion of NEPS.

Outline and results

The main goal of this paper is find the number of solution of the diagonal equation

[TABLE]

over finite fields in a combinatorial way. For this purpose, we relate the number of solution of (1.2) with the number of walks generalized Paley graphs $\Gamma(k,p^{m})=\mathrm{Cay}(\mathbb{F}_{q},R_{k})$ (GP-graph for short). By using a classification of GP-graphs which are NEPS of complete graph due to Lim and Praeger (see [7]), the problem of find the number of solution of (1.2) turns on to calculate the number of walks of NEPS of complete graphs. We will calculate a closed formula for the number of walks in NEPS of complete graphs.

The paper is organized as follows. In section 2, we recall some basic definition of generalized Paley graphs and diagonal equations over finite fields, and will be obtain a direct relationship between the number of $r$-walks from $x$ to $y$ and the number of solution of (1.2) with $\alpha=y-x$, in this case.

In Section 3, we find a closed formula for the number of $r$-walks in NEPS in terms of the number of walks of its factors by using essencially the properties of Kronecker products of matrices and well-known facts about the power of matrices and the number of walks between two vertices.

In Section 4 we apply the formula for the number of walks to the case of the cartesian product of the same complete graph which is $\mathrm{NEPS}(K_{n},\ldots,K_{n};\mathcal{B})$ with $\mathcal{B}$ the cannonical basis. In this case, this graph is the well-known Hamming graph. In [7], the authors characterized those generalized Paley graphs which are Hamming graph. Using this, we find the $r$-walks between two vertices in generalized Paley graphs and thus by aplying the result of section 2 we obtain a formula for the number of solution of the diagonal equation (2.3) over $\mathbb{F}_{p^{m}}$ for $k=\frac{p^{ab}-1}{p^{a}-1}$ where $m=ab$.

2. GP-graphs and diagonal equation over finite fields

Let $p$ be a prime and let $m,k$ be positive integers such that $k\mid p^{m}-1$. The generalized Paley graph is the Cayley graph

[TABLE]

i.e. $\Gamma(k,p^{m})$ is the graph with set of vertex $\mathbb{F}_{p^{m}}$ and two vertices $x,y\in\mathbb{F}_{p^{m}}$ are neighbors if and only if the difference $y-x\in R_{k}$. In general $\Gamma(k,p^{m})$ is a directed graph, but if $R_{k}$ is symmetric ($R_{k}=-R_{k}$), then $\Gamma(k,p^{m})$ is a simple graph.

Notice that if $\omega$ is a primitive element of $\mathbb{F}_{p^{m}}$, then $R_{k}=\langle\omega^{k}\rangle$, this implies that $\Gamma(k,p^{m})$ is a $(\frac{p^{m}-1}{k})$-regular graph. We assume that $u=\frac{p^{m}-1}{k}$ is a primitive divisor of $p^{m}-1$ (i.e. $u$ does not divide $p^{h}-1$ for any $h<m$) and $u$ even if $p$ is odd. The first condition is equivalent to $\Gamma(k,p^{m})$ being a connected graph and the second one is equivalent to $\Gamma(k,p^{m})$ being a simple graph if $p$ is odd. Notice that if $p=2$ then $\Gamma(k,p^{m})$ is a simple graph (without using this condition).

Given a graph $G$ and $v_{i},v_{j}$ vertices of $G$, we denote by $w_{G}(r,v_{i},v_{j})$ to the number of walks of length $r$ from $v_{i}$ to $v_{j}$ in $G$. By convention, $w_{G}(0,v_{i},v_{j})=0$ or $1$ if $v_{i}\neq v_{j}$ or $v_{i}=v_{j}$, respectively. The following Theorem relates the number of solution of the diagonal equations with $\alpha_{i}=1$ and $k_{i}=k$ for all $i=1,\ldots,s$ over $\mathbb{F}_{p^{m}}$ and the walks of the GP-graph $\Gamma(k,p^{m})$.

Theorem 2.1**.**

Let $p$ be a prime and let $k,m$ be positive integers such that $k\mid p^{m}-1$. Given $x,y\in\Gamma(k,p^{m})$, then

[TABLE]

Proof.

If $x,y\in\mathbb{F}_{p^{m}}$, then an $s$-walk from $x$ to $y$ in $\Gamma(k,p^{m})$ gives $x_{1},\ldots,x_{s}\in\mathbb{F}_{p^{m}}^{*}$ such that

[TABLE]

Notice that, given $x\in\mathbb{F}_{p^{m}}^{*}$ there are exactly $k$ elements $y\in\mathbb{F}_{p^{m}}^{*}$ such that $x^{k}=y^{k}$. So, each walk induces $k^{s}$-solutions satisfying (2.3).

Reciprocally, any solution $(x_{1},\ldots,x_{s})\in(\mathbb{F}_{p^{m}}^{*})^{s}$ of (2.3) defines an $s$-walk from $x$ to $y$ in $\Gamma(k,p^{m})$, by taking into account that there are $k$ elements $y\in\mathbb{F}_{p^{m}}^{*}$ such that $x^{k}=y^{k}$ for each $x\in\mathbb{F}_{p^{m}}^{*}$. Thus, there are $k^{s}$ different solutions of (2.3) which induce the same walk. Therefore

[TABLE]

as desired. ∎

Remark 2.2**.**

Notice that the equation (2.2), allow us to obtain the number $M$ of solution of (2.3) in $\mathbb{F}_{p^{m}}^{s}$, by taking into account that

[TABLE]

where $N_{i}$ denotes the number of solution $(x_{1},\ldots,x_{i})\in(\mathbb{F}_{p^{m}}^{*})^{i}$ of $x+x_{1}^{k}+\cdots+x_{i}^{k}=y$. In the case $x=y$, notice that we have the trivial solution $x_{v}=0$ for each $v=1,\ldots,s$, thus we obtain that

[TABLE]

3. Number of walks in NEPS

It is well-known that if $A(G)$ is the adjacency matrix of a graph $G$, then

[TABLE]

labeling the vertices in an appropriate way.

The adjacency matrix of $\mathrm{NEPS}(G_{1},\ldots,G_{n};\mathcal{B})$ can be calculated in terms of the adjacency matrices of the graphs $G_{1},\ldots,G_{n}$. More precisely, if $G=\mathrm{NEPS}(G_{1},\ldots,G_{n};\mathcal{B})$ and the graphs $G_{1},\ldots,G_{n}$ have adjacency matrices $A_{1},\ldots,A_{n}$, then the adjacency matrix of $G$ is given by

[TABLE]

where $\otimes$ denotes the Kronecker matrix product and $\alpha=(\alpha_{1},\ldots,\alpha_{n})$ (see [4]).

Theorem 3.1**.**

If $G=\mathrm{NEPS}(G_{1},\ldots,G_{n},\mathcal{B})$ and if $v_{i},v_{j}\in V(G)$ then

[TABLE]

where $\pi_{t}$ denotes the projection of $V(G)$ in $V(G_{t})$ and $\beta_{\ell}=(\beta_{\ell 1},\beta_{\ell 2},\ldots,\beta_{\ell n})\in\mathcal{B}$ for all $\ell=1,\ldots,r$.

Proof.

Recall that the Kronecker product has the property

[TABLE]

Thus, by (3.2) if $A,A_{1},\ldots,A_{n}$ are the adjacency matrices of $G,G_{1},\ldots,G_{n}$, respectively, then

[TABLE]

where $\beta_{\ell}=(\beta_{\ell 1},\beta_{\ell 2},\ldots,\beta_{\ell n})\in\mathcal{B}$ for all $\ell=1,\ldots,r$ and thus

[TABLE]

Taking into account that $A$ is constructed with the lexicographic order for the vertices of $G$ which represent the ordered $n$-tuples of vertices of $G_{1},\ldots,G_{n}$ (see [4]). Denote by $o_{t}(i)$ the label of $\pi_{t}(v_{i})$ in $G_{t}$ for $t=1,\ldots,n$. By definition of Kronecker product we have

[TABLE]

and, by (3.1)

[TABLE]

Therefore, by (3.1) and (3.4), we obtain the desired formula. ∎

As a consequence we obtain a formula for the number of walks in NEPS of complete graphs.

Corollary 3.2**.**

Let $G=NEPS(K_{m_{1}},\ldots,K_{m_{n}};\mathcal{B})$. If $v_{i},v_{j}\in V(G)$ then

[TABLE]

where

[TABLE]

Proof.

It is enough to find $w_{K_{m}}(r,w_{i},w_{j})$, where $K_{m}$ is the complete graph with $m$ vertices. It is well known that

[TABLE]

Thus, the result follows from Theorem 3.1. ∎

Example 3.3**.**

Let $K_{3}$ and $K_{4}$ be the complete graphs of $3$ and $4$ vertices respectively and let $G_{1}=\mathrm{NEPS}(K_{3}\times K_{4},\mathcal{B}_{1})$ and $G_{2}=\mathrm{NEPS}(K_{3}\times K_{4},\mathcal{B}_{2})$ with $\mathcal{B}_{1}=\{(1,1)\}$ and $\mathcal{B}_{2}=\{(1,0),(0,1)\}$.

Clearly, $\mathcal{B}_{1}^{r}$ only contains the element $\beta=(\beta_{1},\ldots,\beta_{r})$ such that $\beta_{i}=\beta_{j}=(1,1)$ for all $i,j\in\{1,\ldots,r\}$, i.e we have that $\beta_{1t}+\cdots+\beta_{rt}=r$ for $t=1,2$. On the other hand, $\mathcal{B}_{2}^{r}$ contains all the elements $\beta=(\beta_{1},\ldots,\beta_{r})$ such that $\beta_{1t}+\cdots+\beta_{rt}=\ell_{t}$ for $t=1,2$ and $\ell_{i}$’s satisfying $\ell_{1}+\ell_{2}=r$. By Corollary 3.2 we have that

[TABLE]

[TABLE]

for any vertex $v\in V(G)$.

4. Main results

Let $\mathcal{B}=\{e_{1},\ldots,e_{n}\}$, where $e_{i}$ is the $n$-tuple with $1$ in the position $i$ and zeros in the remainning positions. If $G$ is the NEPS of the graphs $G_{1},\ldots,G_{n}$ with basis $\mathcal{B}$, then $G=G_{1}+\cdots+G_{n}$ is the sum of $G_{t}$ (cartesian product of graph). In this case we have the following result.

Proposition 4.1**.**

Let $G=G_{1}+\cdots+G_{n}$. Then, we have that

[TABLE]

where $\pi_{t}$ denote the projection of $V(G)$ over $V(G_{t})$.

Proof.

Let $r$ be a non-negative integer. By Theorem 3.1 we have that

[TABLE]

where $\mathcal{B}=\{e_{1},\ldots,e_{n}\}$. Notice that if $(\beta_{1},\ldots,\beta_{r})\in\mathcal{B}^{r}$, then

[TABLE]

Moreover, there exist a number $\frac{r!}{r_{1}!\cdots r_{n}!}$ of $r$-tuples $(\beta_{1},\ldots,\beta_{r})$’s with $\beta_{1}+\cdots+\beta_{r}=(r_{1},r_{2},\ldots,r_{n})$. Therefore we set (4.1) as we wanted. ∎

Recall that the Hamming graph $H(n,q)$ is the graph with vertex set all the $n$-tuples with entries from a set $\Delta$ of size $q$, and two $n$-tuples are neighbors if and only if they differ in exactly one entry. It is known that $H(n,q)$ is the $n$-sum of the complete graph $K_{q}$. Therefore, we have that

[TABLE]

where $r_{t}\geq 0$ for all $t=1,\ldots,n$ and

[TABLE]

In [7], the authors caracterized all generalized Paley graphs which are Hamming graphs. More precisely, they showed that $\Gamma(k,p^{m})$ is a Hamming graph if and only if $u=b(p^{a}-1)$ for some divisor $b>1$ such that $m=ab$.

Also if $\omega$ is a primitive element of $\mathbb{F}_{p^{m}}$, then the set $\{1,\,\omega^{k},\,\omega^{2k},\ldots,\,\omega^{(b-1)k}\}$ is a basis of $\mathbb{F}_{p^{m}}$ as $\mathbb{F}_{p^{a}}$-vector space, then

[TABLE]

and hence we have the isomorphism (see [7])

[TABLE]

In this case, we have the following result.

Proposition 4.2**.**

Let $p$ be a prime and let $m,k$ be positive integers such that $k\mid p^{m}-1$. If $u=\frac{p^{m}-1}{k}=b(p^{a}-1)$ with $b>1$ and $m=ab$, then

[TABLE]

where $r_{i}\geq 0$ for all $i=1,\ldots,b$ and

[TABLE]

where $[x]_{i}$ denotes the $i$-th coordinate of the vector $[x]\in(\mathbb{F}_{p^{a}})^{b}$, respect to the $\mathbb{F}_{p^{a}}$-base $\{1,\,\omega^{k},\,\omega^{2k},\ldots,\,\omega^{(b-1)k}\}$.

As a direct consequence of the Theorem 2.1 and Proposition 4.2 we obtain the following Proposition.

Proposition 4.3**.**

Let $p$ be a prime and let $a,b$ be positive integers such that $b>1$. If $k=\frac{p^{ab}-1}{b(p^{a}-1)}$ is integer and $N_{r}$ denotes of solutions $(x_{1},\ldots,x_{r})$ in $(\mathbb{F}_{p^{ab}}^{*})^{r}$ to the diagonal equation $x_{1}^{k}+\cdots+x_{r}^{k}=\alpha$, then

[TABLE]

where $r_{i}\geq 0$ for all $i=1,\ldots,b$ and

[TABLE]

where $[\alpha]_{i}$ denotes the $i$-th coordinate of the vector $[\alpha]\in(\mathbb{F}_{p^{a}})^{b}$, respect to the $\mathbb{F}_{p^{a}}$-base $\{1,\,\omega^{k},\,\omega^{2k},\ldots,\,\omega^{(b-1)k}\}$ of $\mathbb{F}_{p^{ab}}$.

As a direct consequence of the last proposition and Remark 2.2 we obtain our main result.

Theorem 4.4**.**

Let $p$ be a prime and let $a,b$ be positive integers such that $b>1$. If $k=\frac{p^{ab}-1}{b(p^{a}-1)}$ is integer and $M_{s}$ denotes of solutions $(x_{1},\ldots,x_{s})$ in $(\mathbb{F}_{p^{ab}})^{s}$ to the diagonal equation $x_{1}^{k}+\cdots+x_{s}^{k}=\alpha$, then

[TABLE]

where $N_{r}$ is given by (4.6).

Remark 4.5**.**

Clearly, the hypothesis $k$ to be integer is equivalent to $b\mid\frac{p^{ab}-1}{p^{a}-1}$. This condition was recently studied in [8]. More specifically, if we put $x=p^{a}$, then $b\mid\frac{p^{ab}-1}{p^{a}-1}$ in the following cases:

1. $(a)$

If $b=r$ is a prime different from $p$ and $x\equiv 1\pmod{r}$. 2. $(b)$

If $b=2r$ with $r$ an odd prime, $x$ coprime with $b$ and $x\equiv\pm 1\pmod{r}$. 3. $(c)$

If $b=rr^{\prime}$ with $r<r^{\prime}$ odd primes such that $r\nmid r^{\prime}-1$ and $x\equiv 1\pmod{rr^{\prime}}$. 4. $(d)$

If $b=r_{1}r_{2}\cdots r_{\ell}$ with $r_{1}<r_{2}<\cdots<r_{\ell}$ primes different from $p$ with $x\equiv 1\pmod{r_{1}}$ and $x^{b/r_{i}}\equiv 1\pmod{r_{i}}$ for $i=2,\ldots,\ell$. 5. $(e)$

If $b=r^{t}$ with $r$ prime such that $ord_{b}(x)=r^{h}$ for some $0\leq h<t$. 6. $(f)$

If $b=r_{1}^{t_{1}}\cdots r_{\ell}^{t_{\ell}}$ with $r_{1}<\cdots<r_{\ell}$ primes different from $p$ where $ord_{r_{i}^{t_{i}}}(x)=r_{i}^{h_{i}}$ with $0\leq h_{i}\leq t_{i}-1$ for all $i$.