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A condition for scattered linearized polynomials involving Dickson matrices

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Abstract

A linearized polynomial over \({{\mathbb {F}}}_{q^n}\) is called scattered when for any \(t,x\in {{\mathbb {F}}}_{q^n}\), the condition \(xf(t)-tf(x)=0\) holds if and only if x and t are \({\mathbb {F}}_q\)-linearly dependent. General conditions for linearized polynomials over \({{\mathbb {F}}}_{q^n}\) to be scattered can be deduced from the recent results in Csajbók (Scalar q-subresultants and Dickson matrices, 2018), Csajbók et al. (Finite Fields Appl 56:109–130, 2019), McGuire and Sheekey (Finite Fields Appl 57:68–91, 2019), Polverino and Zullo (On the number of roots of some linearized polynomials, 2019). Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon–Polverino binomial \(x^{q^s}+\delta x^{q^{n-s}}\), allowing to prove that for any n and s, if \({{\,\mathrm{N}\,}}_{q^n/q}(\delta )=1\), then the binomial is not scattered. Also, a necessary and sufficient condition for \(x^{q^s}+bx^{q^{2s}}\) to be scattered is shown which is stated in terms of a special plane algebraic curve.

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Correspondence to Corrado Zanella.

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Zanella, C. A condition for scattered linearized polynomials involving Dickson matrices. J. Geom. 110, 50 (2019). https://doi.org/10.1007/s00022-019-0505-z

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  • DOI: https://doi.org/10.1007/s00022-019-0505-z

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