This paper presents an explicit representation for the solutions of the equation \({\sum }_{i=0}^{\frac kl-1}x^{2^{li}} = a \in \mathbb {F}_{2^{n}}\) for any given positive integers k, l with l|k and n, in the closed field \({\overline {\mathbb {F}_{2}}}\) and in the finite field \(\mathbb {F}_{2^{n}}\). As a by-product of our study, we are able to completely characterize the a’s for which this equation has solutions in \(\mathbb {F}_{2^{n}}\).

中文翻译：

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解x + x 2 l +⋯+ x 2 ml = a $ x + x ^ {2 ^ {l}} + \ cdots + x ^ {2 ^ {ml}} = a $ F 2 n $ \ mathbb { F} _ {2 ^ {n}} $

本文给出了方程\（{\ sum __ {i = 0} ^ {\ frac kl-1} x ^ {2 ^ {li}} = a \ in \ mathbb {F} _ {2 ^ {n}} \）对于任何给定的正整数k，l与l | k和n，在封闭字段\（{\ overline {\ mathbb {F} _ {2}}} \\）和有限字段\（\ mathbb {F} _ {2 ^ {n}} \\）中。作为我们研究的副产品，我们能够完全刻画该方程在\（\ mathbb {F} _ {2 ^ {n}} \}中具有解的a的特征。