The graph of a family of functions over quadratic extensions of finite fields

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-04-01 DOI:10.1016/j.disc.2025.114500

Claude Gravel , Daniel Panario , Hugo Teixeira

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引用次数: 0

Abstract

Brochero and Teixeira (2023) [4] showed the behavior of the functional graph of

f_{a} (X) = X^{q + 1} + a X^{2}

over quadratic extensions of finite fields explicitly for

a \in {1, - 1}

. In this article, we create a family of functions using repeated iterations of the function

f_{a} (X)

and taking values of

a \in {1, - 1}

in each iteration. Let α be an n-sequence of values for a, taken over

{1, - 1}

, and

f_{α} (X)

be the resulting function. We present the form of

f_{α} (X)

and use it to derive a closed formula for the number and length of cycles present in the functional graph of

f_{α} (X)

. We then determine the shape of the trees hanging from each cycle and gather all the results in our main theorem.

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.