Geodesic
On the number of solutions of the equation $(x_1+\ldots+x_n)^m=ax_1\ldots x_n$ in a finite field
Diskretnaya Matematika, Tome 16 (2004) no. 4, pp. 41-48

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We consider the equation $(x_1+\ldots +x_n)^m=ax_1\ldots x_n$, where $a$ is a nonzero element of the finite field $\mathbf F_q$, $n\ge 2$, and $m$ is a positive integer. Explicit formulas for the number of solutions of this equation in $\mathbf F_q^n$ under the condition $d\in\{1,2,3,6\}$, where $d=\mathrm{gcd}(m-n,q-1)$, are found. Moreover, we obtain formulas for the number of solutions for arbitrary $d>2$ if there exists positive integer $l$ such that $d\mid(p^l+1)$, where $p$ is the characteristic of $\mathbf F_q$. 
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     author = {Yu. N. Baulina},
     title = {On the number of solutions of the equation $(x_1+\ldots+x_n)^m=ax_1\ldots x_n$ in a finite field},
     journal = {Diskretnaya Matematika},
     pages = {41--48},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2004_16_4_a4/}
}
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Yu. N. Baulina. On the number of solutions of the equation $(x_1+\ldots+x_n)^m=ax_1\ldots x_n$ in a finite field. Diskretnaya Matematika, Tome 16 (2004) no. 4, pp. 41-48. http://geodesic.mathdoc.fr/item/DM_2004_16_4_a4/