Iterating additive polynomials over finite fields

被引:0
作者
Reis, Lucas [1 ]
机构
[1] Univ Fed Minas Gerais, Dept Matemat, Belo Horizonte, MG, Brazil
关键词
dynamics over finite fields; additive polynomials; factorization of polynomials; PERIODIC POINTS;
D O I
10.1017/S0013091525000173
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let q be a power of a prime p, let $\mathbb F_q$ be the finite field with q elements and, for each nonconstant polynomial $F\in \mathbb F_{q}[X]$ and each integer $n\ge 1$, let $s_F(n)$ be the degree of the splitting field (over $\mathbb F_q$) of the iterated polynomial $F<^>{(n)}(X)$. In 1999, Odoni proved that $s_A(n)$ grows linearly with respect to n if $A\in \mathbb F_q[X]$ is an additive polynomial not of the form $aX<^>{p<^>h}$; moreover, if q = p and $B(X)=X<^>p-X$, he obtained the formula $s_{B}(n)=p<^>{\lceil \log_p n\rceil}$. In this paper we note that $s_F(n)$ grows at least linearly unless $F\in \mathbb F_q[X]$ has an exceptional form and we obtain a stronger form of Odoni's result, extending it to affine polynomials. In particular, we prove that if A is additive, then $s_A(n)$ resembles the step function $p<^>{\lceil \log_p n\rceil}$ and we indeed have the identity $s_A(n)=\alpha p<^>{\lceil \log_p \beta n\rceil}$ for some $\alpha, \beta\in \mathbb Q$, unless A presents a special irregularity of dynamical flavour. As applications of our main result, we obtain statistics for periodic points of linear maps over $\mathbb F_{q<^>i}$ as $i\to +\infty$ and for the factorization of iterates of affine polynomials over finite fields.
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页数:16
相关论文
共 25 条
[1]   ON STABLE QUADRATIC POLYNOMIALS [J].
Ahmadi, Omran ;
Luca, Florian ;
Ostafe, Alina ;
Shparlinski, Igor E. .
GLASGOW MATHEMATICAL JOURNAL, 2012, 54 (02) :359-369
[2]   Stability of polynomials [J].
Ali, N .
ACTA ARITHMETICA, 2005, 119 (01) :53-63
[3]   Irreducibility of the iterates of a quadratic polynomial over a field [J].
Ayad, M ;
McQuillan, DL .
ACTA ARITHMETICA, 2000, 93 (01) :87-97
[4]   On the number of distinct functional graphs of affine-linear transformations over finite fields [J].
Bach, Eric ;
Bridy, Andrew .
LINEAR ALGEBRA AND ITS APPLICATIONS, 2013, 439 (05) :1312-1320
[5]   CURRENT TRENDS AND OPEN PROBLEMS IN ARITHMETIC DYNAMICS [J].
Benedetto, Robert ;
Ingram, Patrick ;
Jones, Rafe ;
Manes, Michelle ;
Silverman, Joseph H. ;
Tucker, Thomas J. .
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, 2019, 56 (04) :611-685
[6]   Iterated monodromy groups of rational functions and periodic points over finite fields [J].
Bridy, Andrew ;
Jones, Rafe ;
Kelsey, Gregory ;
Lodge, Russell .
MATHEMATISCHE ANNALEN, 2024, 390 (01) :439-475
[7]   On the irreducibility of the iterates of xn-b [J].
Danielson, L ;
Fein, B .
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2002, 130 (06) :1589-1596
[8]  
Elspas B., 1959, IRE Trans. Circ. Theory CT, V6, P45, DOI DOI 10.1109/TCT.1959.1086506
[9]   PERIODIC POINTS OF POLYNOMIALS OVER FINITE FIELDS [J].
Garton, Derek .
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 2022, 375 (07) :4849-4871
[10]   On irreducible divisors of iterated polynomials [J].
Gomez-Perez, Domingo ;
Ostafe, Alina ;
Shparlinski, Igor E. .
REVISTA MATEMATICA IBEROAMERICANA, 2014, 30 (04) :1123-1134