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Solving $X^{2^{3n}+2^{2n}+2^{n}-1}+(X+1)^{2^{3n}+2^{2n}+2^{n}-1}=b$ in $GF{2^{4n}}$

This article determines all the solutions in the finite field $GF{2^{4n}}$ of the equation $x^{2^{3n}+2^{2n}+2^{n}-1}+(x+1)^{2^{3n}+2^{2n}+2^{n}-1}=b$. Specifically, we explicitly determine the set of $b$'s for which the equation has $i$ solutions for any positive integer $i$. Such sets, which depend on the number of solutions $i$, are given explicitly and expressed nicely, employing the absolute trace function over $GF{2^{n}}$, the norm function over $GF{2^{4n}}$ relatively to $GF{2^{n}}$ and the set of $2^n+1$st roots of unity in $GF{2^{4n}}$. The equation considered in this paper comes from an article by Budaghyan et al. \cite{BCCDK20}. As an immediate consequence of our results, we prove that the above equation has $2^{2n}$ solutions for one value of $b$, $2^{2n}-2^n$ solutions for $2^n$ values of $b$ in $GF{2^{4n}}$ and has at most two solutions for all remaining points $b$, leading to complete proof of the conjecture raised by Budaghyan et al. We highlight that the recent work of Li et al., in \cite{Li-et-al-2020} gives the complete differential spectrum of $F$ and also gives an affirmative answer to the conjecture of Budaghyan et al. However, we emphasize that our approach is interesting and promising by being different from Li et al. Indeed, on the opposite to their article, our technique allows determine ultimately the set of $b$'s for which the considered equation has solutions as well as the solutions of the equation for any $b$ in $GF{2^{4n}}$.