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Prikladnaya Diskretnaya Matematika. Supplement, 2022, Issue 15, Pages 5–8
DOI: https://doi.org/10.17223/2226308X/15/1
(Mi pdma566)
 

This article is cited in 3 scientific papers (total in 3 papers)

Theoretical Foundations of Applied Discrete Mathematics

Invariant subspaces of functions affine equivalent to the finite field inversion

N. A. Kolomeetsa, D. A. Bykovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
Full-text PDF (628 kB) Citations (3)
References:
Abstract: In the paper, we consider affine $\mathbb{F}_{p}$-subspaces of a finite field $\mathbb{F}_{p^n}$, $p$ is prime, such that the function $x^{-1}$ which inverses a field element $x$ (we assume that $0^{-1}$ = 0) maps them to affine subspaces. It is proven that the image of an affine subspace $U$, $|U| > 2$, is an affine subspace as well if and only if $U = q \mathbb{F}_{p^k}$, where $q \in \mathbb{F}^*_{p^n}$ and $k | n$. In other words, these subspaces can be expressed using subfields of $\mathbb{F}_{p^n}$. As a consequence, we propose a sufficent condition providing that a function $A(x^{-1}) + b$ has no invariant affine subspaces $U$ of cardinality $2 < |U| < p^n$, where $A: \mathbb{F}_{p^n} \to \mathbb{F}_{p^n}$ is an invertible $\mathbb{F}_{p}$-linear transformation, $b \in \mathbb{F}^*_{p^n}$. Also, we give examples of functions which have no invariant affine subspaces except for $\mathbb{F}_{p^n}$.
Keywords: finite fields, inversion, affine subspaces, invariant subspaces.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation FWNF-2022-0018
Document Type: Article
UDC: 519.7
Language: Russian
Citation: N. A. Kolomeets, D. A. Bykov, “Invariant subspaces of functions affine equivalent to the finite field inversion”, Prikl. Diskr. Mat. Suppl., 2022, no. 15, 5–8
Citation in format AMSBIB
\Bibitem{KolByk22}
\by N.~A.~Kolomeets, D.~A.~Bykov
\paper Invariant subspaces of functions affine equivalent to the finite field inversion
\jour Prikl. Diskr. Mat. Suppl.
\yr 2022
\issue 15
\pages 5--8
\mathnet{http://mi.mathnet.ru/pdma566}
\crossref{https://doi.org/10.17223/2226308X/15/1}
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Prikladnaya Diskretnaya Matematika. Supplement
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    Full-text PDF :36
    References:13
     
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