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.
Similar content being viewed by others
Notes
References
Bartoli, D., Giulietti, M., Marino, G., Polverino, O.: Maximum scattered linear sets and complete caps in Galois spaces. Combinatorica 38, 255–278 (2018)
Bartoli, D., Zhou, Y.: Exceptional scattered polynomials. J. Algebra 509, 507–534 (2018)
Blokhuis, A., Lavrauw, M.: Scattered spaces with respect to a spread in \({\rm PG}(n, q)\). Geom. Dedicata 81, 231–243 (2000)
Csajbók: Scalar \(q\)-subresultants and Dickson matrices (2019). arXiv:1909.06409
Csajbók, B., Marino, G., Polverino, O., Zanella, C.: A new family of MRD-codes. Linear Algebra Appl. 548, 203–220 (2018)
Csajbók, B., Marino, G., Zullo, F.: New maximum scattered linear sets of the projective line. Finite Fields Appl. 54, 133–150 (2018)
Csajbók, B., Marino, G., Polverino, O., Zullo, F.: A characterization of linearized polynomials with maximum kernel. Finite Fields Appl. 56, 109–130 (2019)
Csajbók, B., Zanella, C.: On the equivalence of linear sets. Des. Codes Cryptogr. 81, 269–281 (2016)
Csajbók, B., Zanella, C.: On scattered linear sets of pseudoregulus type in \({\rm PG}(1, q^t)\). Finite Fields Appl. 41, 34–54 (2016)
Csajbók, B., Zanella, C.: Maximum scattered \({\mathbb{F}}_q\)-linear sets of \({\rm PG}(1, q^4)\). Discrete Math. 341, 74–80 (2018)
Lavrauw, M., Marino, G., Polverino, O., Trombetti, R.: Solution to an isotopism question concerning rank 2 semifields. J. Combin. Des. 23, 60–77 (2015)
Lunardon, G., Polverino, O.: Blocking sets and derivable partial spreads. J. Algebr. Combin. 14, 49–56 (2001)
Lunardon, G., Trombetti, R., Zhou, Y.: Generalized twisted Gabidulin codes. J. Combin. Theory Ser. A 159, 79–106 (2018)
Marino, G., Montanucci, M., Zullo, F.: MRD-codes arising from the trinomial \(x^q+x^{q^3}+cx^{q^5}\in {\mathbb{F}}_{q^6}[x]\) (2019). arXiv:1907.08122
McGuire, G., Sheekey, J.: A characterization of the number of roots of linearized and projective polynomials in the field of coefficients. Finite Fields Appl. 57, 68–91 (2019)
Montanucci, M.: Private communication (2019) (in progress)
Montanucci, M., Zanella, C.: A class of linear sets in \({\rm PG} (1, q^5)\) (2019). arXiv:1905.10772
Polverino, O.: Linear sets in finite projective spaces. Discrete Math. 310, 3096–3107 (2010)
Polverino, O., Zullo, F.: On the number of roots of some linearized polynomials (2019). arXiv:1909.00802
Sheekey, J.: A new family of linear maximum rank distance codes. Adv. Math. Commun. 10(3), 475–488 (2016)
Wu, B., Liu, Z.: Linearized polynomials over finite fields revisited. Finite Fields Appl. 22, 79–100 (2013)
Zanella, C., Zullo, F.: Vertex properties of maximum scattered linear sets of \({\rm PG} (1,q^n)\) (2019). arXiv:1906.05611
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00022-019-0505-z