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Solving $$x+x^{2^l}+\ldots +x^{2^{ml}}=a$$ over $$\mathbb{F}_{2^n}$$. (English) Zbl 1446.11207
Summary: This paper presents an explicit representation for the solutions of the equation $$\sum_{i=0}^{\frac{k}{l}-1} x^{2^{li}} = a \in \mathbb{F}_{2^n}$$ for any given positive integers $$k, l$$ with $$l\mid k$$ 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 $$\mathbb{F}_{2^n}$$.

##### MSC:
 11T06 Polynomials over finite fields 12E05 Polynomials in general fields (irreducibility, etc.)
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##### References:
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