arXiv subjects

Yuanpeng Zeng

Publications and source records attributed to Yuanpeng Zeng.

3 recordsLinked to original sources

arXiv2026

On cyclically covering subspaces of $\mathbb{F}^n_q$

For a prime power $ q $ and a positive integer $ n $, a subspace $ U \subseteq \mathbb{F}_q^n $ is called cyclically covering if the union of all its cyclic shifts covers the whole space $ \mathbb{F}_q^n $. Let $ h_q(n) $ denote the maximum possible codimension of such a subspace. This paper focuses on the case $ h_q(n) = 0 $. We provide necessary and sufficient conditions under which $ h_q(n) = 0 $ holds. As an application, we show that $ h_q(\ell^t) = 0 $ whenever $ q $ is a primitive root modulo $ \ell^t $. Moreover, we prove that if $ n $ is odd and $ h_q(n) = 0 $, then also $ h_q(2n) = 0 $. As an example, we show that $ h_3(11) =h_3(16) = 1 $. Furthermore, we investigate the relationship between the coverings of $\mathbb{F}_{q^m}^n$ and $\mathbb{F}_q^{mn}$, and obtain several sufficient conditions for $h_{q^m}(n) = 0$. Specifically, we derive that if $n = 3$ or $n = 2^d$ (where $d$ is a nonnegative integer), then $h_4(n) = 0$.

math.NT↗

arXiv2025

Determination of Some Types of Permutations over $\mathbb{F}_q^2$ with Low-Degree

The characterization of permutations over finite fields is an important topic in number theory with a long-standing history. This paper presents a systematic investigation of low-degree bivariate polynomial systems $F=(f_1(x,y),f_2(x,y))$ defined over $\mathbb{F}_{q}^2$. Specifically, we employ Hermite's Criterion to completely classify bivariate quadratic permutation polynomial systems, while utilizing the theory of permutation rational functions to give a full classification of bivariate 3-homogeneous permutation polynomial systems. Furthermore, as an application of our findings, we provide an explicit characterization of the permutation binomials of the form $x^3+ax^{2q+1}$ over $\mathbb{F}_{q^2}$ with characteristic $p\neq3$, thereby resolving a significant special case within this classical research domain.

math.NT↗

arXiv2025

Permutation polynomials of the form $x+γ\mathrm{Tr}(H(x))$

Given a polynomial $ H(x) $ over $\mathbb{F}_{q^n}$, we study permutation polynomials of the form $ x + γ\mathrm{Tr}(H(x)) $ over $\mathbb{F}_{q^n}$. Let \[P_H=\{γ\in \mathbb{F}_{q^n} : x+γ\mathrm{Tr}(H(x))~\text{is a permutation polynomial}\}.\] We present some properties of the set $P_H$, particularly its relationship with linear translators. Moreover, we obtain an effective upper bound for the cardinality of the set $P_H$ and show that the upper bound can reach up to $q^n - q^{n - 1}$. Furthermore, we prove that when the cardinality of the set $P_H$ reaches this upper bound, the function $\mathrm{Tr}(H(x))$ must be an $\mathbb{F}_q$-linear function. Finally, we study two classes of functions $H(x)$ over $\mathbb{F}_{q^2}$ and determine the corresponding sets $P_H$. The sizes of these sets $P_H$ are all relatively small, even only including the trivial case.

math.NT↗