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Pingzhi Yuan

Publications and source records attributed to Pingzhi Yuan.

At least 37 records · Page 2Linked to original sources
arXiv2025

Algebraic Structure of Permutational Polynomials over $\mathbb{F}_{q^n}$ \uppercase\expandafter{\romannumeral2}

It is well known that there exists a significant equivalence between the vector space $\mathbb{F}_{q}^n$ and the finite fields $\mathbb{F}_{q^n}$, and many scholars often view them as the same in most contexts. However, the precise connections between them still remain mysterious. In this paper, we first show their connections from an algebraic perspective, and then propose a more general algebraic framework theorem. Furthermore, as an application of this generalized algebraic structure, we give some classes of permutation polynomials over $\mathbb{F}_{q^2}$.

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arXiv2025

Fermat's and Catalan's equations over $M_2(\mathbb{Z})$

Let $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right)$ be a given matrix such that $bc\neq0$ and let $C(A)=\{B\in M_2(\mathbb{Z}): AB=BA\}$. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $uX^i+vY^j=wZ^k,\, i,\, j,\, k\in\mathbb{N},\, X, \,Y,\, Z\in C(A)$, where $u,\, v,\, w$ are given nonzero integers such that $\gcd\left(u,\, v,\, w\right)=1$. From this, we get a necessary and sufficient condition for the solvability of the Fermat's matrix equation in $C(A)$. Moreover, we show that the solvability of the Catalan's matrix equation in $M_2\left(\mathbb{Z}\right)$ can be reduced to the solvability of the Catalan's matrix equation in $C(A)$, and finally to the solvability of the Catalan's equation in quadratic fields.

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arXiv2025

Some permutation polynomials via linear translators

Permutation polynomials with explicit constructions over finite fields have long been a topic of great interest in number theory. In recent years, by applying linear translators of functions from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, many scholars constructed some classes of permutation polynomials. Motivated by previous works, we first naturally extend the notion of linear translators and then construct some permutation polynomials.

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arXiv2025

Faithful Decomposition of Rationals

If an irreducible fraction $\frac mn>0$ can be decomposed into the sum of several irreducible proper fractions with different denominators, and the positive number smaller than $\frac mn$ in fractional ideal $\frac 1n\mathbb Z$ can not be obtained by replacing some numerator with smaller non-negative integers, then the decomposition is said to be faithful. For $t\in\mathbb Z$, we prove that the length of faithful decomposition of an irreducible fraction $\frac mn$ with $2\le t\le\frac mn<t+1$ is at least $t+2$. In addition, we show a faithful decomposition of rationals consisting only of unit fractions except for one term. And we write $\frac 4n$ as a faithful decomposition with three fractions at most one non-unit fraction.

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arXiv2025

On the representation of rational numbers via Euler's totient function

Let $b>1$ be an odd positive integer and $k, l \in \mathbb{N}$. In this paper, we show that every positive rational number can be written as $φ(m^{2})/(φ(n^{2}))^{b}$ and $φ(k(m^{2}-1))/φ(ln^{2})$, where $m, n\in \mathbb{N}$ and $φ$ is the Euler's totient function. At the end, some further results are discussed.

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arXiv2025

On the Elementary Symmetric Functions of $\{1,1/2,\dots,1/n\}\backslash\{1/i\}$

In 1946, P. Erdős and I. Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1,1 / 2$, $\cdots, 1 / n$ are integers. In 2012, Y. Chen and M. Tang proved that if $n \geqslant 4$, then none of the elementary symmetric functions of $1,1 / 2, \cdots, 1 / n$ are integers. In this paper, we prove that if $n \geqslant 5$, then none of the elementary symmetric functions of $\{1,1 / 2, \cdots, 1 / n\} \backslash\{1 / i\}$ are integers except for $n=i=2$ and $n=i=4$.

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arXiv2024

The compositional inverses of three classes of permutation polynomials over finite fields

Recently, P. Yuan presented a local method to find permutation polynomials and their compositional inverses over finite fields. The work of P. Yuan inspires us to compute the compositional inverses of three classes of the permutation polynomials: (a) the permutation polynomials of the form $ax^q+bx+(x^q-x)^k$ over $\mathbb{F}_{q^2},$ where $a+b \in \mathbb{F}_q^*$ or $a^q=b;$ (b) the permutation polynomials of the forms $f(x)=-x+x^{(q^2+1)/2}+x^{(q^3+q)/2} $ and $f(x)+x$ over $\mathbb{F}_{q^3};$ (c) the permutation polynomial of the form $A^{m}(x)+L(x)$ over $\mathbb{F}_{q^n},$ where ${\rm Im}(A(x))$ is a vector space with dimension $1$ over $\mathbb{F}_{q}$ and $L(x)$ is not a linearized permutation polynomial.

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arXiv2024

The compositional inverses of permutation polynomials from trace functions over finite fields

In this paper, we present the compositional inverses of several classes permutation polynomials of the form $\sum_{i=1}^kb_i\left({\rm Tr}_m^{mn}(x)^{t_i}+δ\right)^{s_i}+f_1(x)$, where $1\leq i \leq k,$ $s_i$ are positive integers, $b_i \in \mathbb{F}_{p^m},$ $p$ is a prime and $f_1(x)$ is a polynomial over $\mathbb{F}_{p^{mn}}$ satisfying the following conditions: (i) ${\rm Tr}_m^{mn}(x) \circ f_1(x)=φ(x) \circ {\rm Tr}_m^{mn}(x),$ where $φ(x)$ is a polynomial over $\mathbb{F}_{p^m};$ (ii) For any $a \in \mathbb{F}_{p^m},$ $f_1(x)$ is injective on ${\rm Tr}_m^{mn}(a)^{-1}.$

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arXiv2024

Permutation polynomials over finite fields by the local criterion

In this paper, we further investigate the local criterion and present a class of permutation polynomials and their compositional inverses over $ \mathbb{F}_{q^2}$. Additionally, we demonstrate that linearized polynomial over $\mathbb{F}_{q^n}$ is a local permutation polynomial with respect to all linear transformations from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q ,$ and that every permutation polynomial is a local permutation polynomial with respect to certain mappings.

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arXiv2022

The matrix equation $aX^m+bY^n=cI$ over $M_2(\mathbb{Z})$

Let $\mathbb{N}$ be the set of all positive integers and let $a,\, b,\, c$ be nonzero integers such that $\gcd\left(a,\, b,\, c\right)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation $aX^m+bY^n=cI,\,X,\,Y\in M_2(\mathbb{Z}),\, m,\, n\in\mathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XY\neq YX$ and the solvability of the equation $ax^m+by^n=c,\, m,\, n\in\mathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^nI,\,X,\,Y\in M_2(\mathbb{Z}),\,n\in\mathbb{N},\,n\geq3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,\, n\in\mathbb{N},\,n\geq3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $aX^2+bY^2=cI,\,X,\,Y\in M_2(\mathbb{Z})$.

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arXiv2022

Positive rational number of the form $φ(km^{a})/φ(ln^{b})$

Let $k, l, a$ and $b$ be positive integers with $\max\{a, \, b\}\ge2$. In this paper, we show that every positive rational number can be written as the form $φ(km^{a})/φ(ln^{b})$, where $m, \, n\in\mathbb{N}$ if and only if $\gcd(a, \,b)=1$ or $(a, b, k, l)=(2,2, 1, 1)$. Moreover, if $\gcd(a, b)>1$, then the proper representation of such representation is unique.

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arXiv2022

Permutation trinomials over $\mathbb{F}_{2^m}$: a corrected version

Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we determine all permutation trinomials over $\mathbb{F}_{2^m}$ in Zieve's paper. We prove a conjecture proposed by Gupta and Sharma and obtain some new permutation trinomials over $\mathbb{F}_{2^m}$. Finally, we show that some classes of permutation trinomials with parameters are QM equivalent to some known permutation trinomials.

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arXiv2022

Permutation Polynomials and their Compositional Inverses

In this paper, we prove that every PP is an AGW-PP. We also extend the result of Wan and Lidl to other permutation polynomials over finite fields and determine their group structure. Moreover, we provide a new general method to find the compositional inverses of all PPs, some new PPs and their compositional inverses are given.

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