Recent zbMATH articles in MSC 12Ehttps://www.zbmath.org/atom/cc/12E2021-04-16T16:22:00+00:00WerkzeugOn a type of permutation rational functions over finite fields.https://www.zbmath.org/1456.112192021-04-16T16:22:00+00:00"Hou, Xiang-dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dong"Sze, Christopher"https://www.zbmath.org/authors/?q=ai:sze.christopherSummary: Let \(p\) be a prime and \(n\) be a positive integer. Let \(f_b(X)=X+(X^p-X+b)^{-1}\), where \(b\in\mathbb{F}_{p^n}\) is such that \(\text{Tr}_{p^n/p}(b)\neq 0\). In 2008, \textit{J. Yuan} et al. [Finite Fields Appl. 14, No. 2, 482--493 (2008; Zbl 1211.11136)] showed that for \(p=2,3,f_b\) permutes \(\mathbb{F}_{p^n}\) for all \(n\geq 1\). Using the Hasse-Weil bound, we show that when \(p>3\) and \(n\geq 5, f_b\) does not permute \(\mathbb{F}_{p^n} \). For \(p>3\) and \(n=2\), we prove that \(f_b\) permutes \(\mathbb{F}_{p^2}\) if and only if \(\text{Tr}_{p^2/p}(b)=\pm 1\). We conjecture that for \(p>3\) and \(n=3,4,f_b\) does not permute \(\mathbb{F}_{p^n}\).Irreducible factors of a class of permutation polynomials.https://www.zbmath.org/1456.112292021-04-16T16:22:00+00:00"Kalaycı, Tekgül"https://www.zbmath.org/authors/?q=ai:kalayci.tekgul"Stichtenoth, Henning"https://www.zbmath.org/authors/?q=ai:stichtenoth.henning"Topuzoğlu, Alev"https://www.zbmath.org/authors/?q=ai:topuzoglu.alevThe authors present results on the degrees of the irreducible factors of a permutation polynomial of the finite field \({\mathbb F}_q\) of \(q\) elements.
More precisely, they study permutation polynomials \(f_n(x)\) recursively defined by
\[f_0(x)=ax+a_0,\quad f_i(x)=f_{i-1}^{d_i}(x)+a_i,~i=1,\ldots,n,\]
where \(a\in {\mathbb F}_q^*\), \(a_0,\ldots,a_n\in {\mathbb F}_q\) and \(d_1,\ldots,d_n\ge 2\) with \(\gcd(d_i,q-1)=1\), \(i=1,\ldots,n\).
Note that each permutation polynomial is of this form with \(d_i=q-2\), \(i=1,\ldots,n\), and some \(n\) by a well-known result of \textit{L. Carlitz} [Proc. Am. Math. Soc. 11, 456--459 (1960; Zbl 0095.03003)].
The first main result (Theorem 2.2) is the following. Let \(d=\mathrm{lcm}(d_1,\ldots,d_n)\) and assume \(\gcd(d,q)=1\). Then the degree of each irreducible factor \(Q(x)\) of \(F_n(x)\) is a divisor of \(d_1 d_2 \cdots d_{n-1}\mathrm{ord}_d(q)\).
Moreover, (Theorem 2.3) the degree of \(Q(x)\) is either \(1\) or divisible by ord\(_\ell(q)\) for some prime divisor \(\ell\) of \(d\). Several additional results are proved.
The results of this paper enable one to produce families of permutation polynomials of large degrees, where possible degrees of their irreducible factors are known. See also the follow-up paper of the same authors [ `` Permutation polynomials and factorization'', Cryptogr. Commun. 12, No. 5, 913--934 (2020; \url{doi:10.1007/s12095-020-00446-y})], in particular, in view of applications.
Reviewer: Arne Winterhof (Linz)On Brauer \(p\)-dimensions and absolute Brauer \(p\)-dimensions of Henselian fields.https://www.zbmath.org/1456.160152021-04-16T16:22:00+00:00"Chipchakov, Ivan D."https://www.zbmath.org/authors/?q=ai:chipchakov.ivan-dLet \(E\) be a field. For each prime number \(p\), let \(Brd_p(E)\) be the Brauer \(p\)-dimension of \(E\) and \(abrd_p(E)\) be the absolute Brauer \(p\)-dimension of \(E\). In this paper, the author studies the sequences \(Brd_p(E)\) and \(abrd_p(E)\), when \(p\) runs over the set \(\mathbb{P}\) of the prime numbers. Suppose that \((E,v)\) is a Henselian valued field with residue field \(\hat E\). For \(p\in\mathbb{P}\), under some restrictions on \(\hat E\), such as \(abrd_p(\hat E)=0\), the author determines
\(Brd_p(E)\) and \(abrd_p(E)\) if \(\mathrm{char}(\hat E)\not=p\). Let \(\Sigma_0\) be the set of sequences \(Brd_p(E)\) and \(abrd_p(E)\) where \(p\in \mathbb{P}\) and \(E\) runs across the class of Henselian valued fields with \(\mathrm{char}(\hat E)\not=0\) and a projective absolute Galois group \(\mathcal G_{\hat E}\). The author gives a description for \(\Sigma_0\). Especially, \(\Sigma_0\) admits a sequence \(a_p,b_p\in\mathbb{N}\cup\{0,\infty\}\), \(p\in \mathbb{P}\), with \(a_2\leq 2b_2\) and
\(a_p\geq b_p\). The paper covers similar results in the case of nonzero characteristic.
Reviewer: Ali Benhissi (Monastir)