arXiv subjects

Pingzhi Yuan

Publications and source records attributed to Pingzhi Yuan.

At least 19 recordsLinked to original sources

arXiv2026

Critical Restricted Sumsets at the Boundary $\lvert A\rvert+\lvert B\rvert=p$

Let $p$ be an odd prime, and write $A\RS B=\{a+b:a\in A,\ b\in B,\ a\ne b\}$. We classify all pairs $A,B\subseteq\Fp$ satisfying $|A|+|B|=p$, $|A|>|B|=\ell\ge1$, and $|A\RS B|=p-2$. After normalizing the two missing sums to $\{0,1\}$, we obtain exactly $\ell+1$ explicit models. The classification forces $B\subseteq A$ and yields, for fixed $p,\ell$, exactly $\binom p2(\ell+1)$ ordered pairs and $1+\lfloor\ell/2\rfloor$ equivalence classes under simultaneous affine transformations. We determine when either set is an arithmetic progression and compute exact difference-set cardinalities. For $\ell\ge3$ and $p\ge4\ell-5$, the cardinality $|B-B|$ determines the equivalence class at fixed $p,\ell$. The proof uses punctured translates and cyclic component counting. We also exhibit a critical pair in $\mathbb F_{13}$, with size gap three and $|A|+|B|=p-2$, that contradicts a proposed inverse statement below the boundary.

math.NT↗

arXiv2026

A Multiplicative Fourier Proof of the Length-Four Index Conjecture

Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157φ(n)/1440<φ(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.

math.NT↗

arXiv2026

On many-to-one mappings over finite fields

We introduce the definition of $m$-to-$1$ mappings between two finite sets, which unifies and generalizes the definitions of $2$-to-$1$ and $n$-to-$1$ mappings in recent literature. We also characterize these $m$-to-$1$ mappings in terms of the generalized local criterion and thus provide three generic constructions of $m$-to-$1$ mappings, which unify and generalize the previous known constructions. Using these constructions, the problem whether $x^r h(x^s)$ is $m$-to-$1$ on the multiplicative group $\mathbb{F}_{q}^{*}$ is converted into that whether an associated polynomial $x^{r_1} h(x)^{s_1}$ is $m_2$-to-$1$ on the order~$\ell$ subgroup~$U_{\ell}$ of $\mathbb{F}_{q}^{*}$, where $m_2 = m / (r, s)$ and $\ell = (q-1) / s$. Furthermore, the $m_2$-to-$1$ property of $x^{r_1} h(x)^{s_1}$ on $U_{\ell}$ is studied in detail in four different cases. In addition, a recursive construction of $m$-to-$1$ mappings from $m$-to-$1$ mappings is proposed.

cs.IT↗

arXiv2026

Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes

Let $g$ be a monic polynomial of degree $r<n$, and let $C_g(n)$ be the coefficient-vector code formed by multiples $ug$ with $°(ug)<n$. We study the coefficient-space MDS locus $M_{n,r}$. The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of $\operatorname{Gr}(r,n)$. We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient $A_0$ times a Schur polynomial $S_κ(g)=s_κ(Λ_g)$, where $κ\subseteq (n-r)^{r-1}$. Hence the universal MDS polynomial is \[D_{n,r}=A_0\prod_{κ\subseteq (n-r)^{r-1}}S_κ.\] This description yields a flat non-MDS boundary over $\mathbb{Z}$, explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for $r\ge 3$ and $N\ge r+3$, every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.

cs.IT↗

arXiv2026

Discrete Fourier Transform Approach to Cyclically Covering Subspaces of $\mathbb{F}^n_q$

Let $q$ be a prime power and $n$ a positive integer. 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. When $\gcd(q,n)=1$, we derive necessary and sufficient conditions for $h_q(n)=0$ via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of $\mathbb{F}_q^n$. We also characterize codimension-$k$ cyclically covering subspaces. {Under suitable coprimality conditions on $m,n$ and on the multiplicative orders of $q$, we prove that the vanishing of $h_q(m)$ and $h_q(n)$ is preserved under their product.} Based on these results, we give a unified characterization of $h_q(n)$ in the case where $q$ and $n$ are primes with $n>q$ and $q$ being a primitive root modulo $n$. Specifically, $h_2(n) \geq 2$ and $h_q(n) = 0$ for $q \neq 2$. We prove that $h_3(n) \ge 1$ for every prime $n > 3$ with odd $\operatorname{ord}_n(3)$. Moreover, for any prime $q > 3$, the Generalized Riemann Hypothesis implies the existence of infinitely many primes $n > q$ such that $q$ is not a primitive root modulo $n$ and $h_q(n) = 0$. We provide algebraic interpretations for the inequalities $h_q(mn)\ge\max\{h_q(m),h_q(n)\}$ and $h_q(mn)\ge h_q(m)+h_q(n)$. Using Galois descent, we prove $h_{q^m}(n)\le h_q(n)$. Furthermore, we generalize a class of constructions that achieve the upper bound $\lfloor\log_q(n)\rfloor$. Finally, under the Generalized Riemann Hypothesis, we obtain average lower bounds of $h_q(n)$ for $q=2,3$.

math.NT↗

arXiv2026

Cyclic Projective Orbits on Rational Normal Curves and MDS Codes

Let $A$ be a cyclic operator on an $r$-dimensional vector space over a field $k$, and let $z$ be a cyclic vector. Their Krylov code has parity-check matrix $(z,Az,\ldots,A^{n-1}z)$. For $r\ge 3$ and $n\ge r+3$, we prove that an MDS orbit segment lies on a rational normal curve precisely when the projective pair $(A,[z])$ is conjugate to one arising from the $(r-1)$-st symmetric-power action of $\mathrm{PGL}_2$. Over finite fields, for companion operators, this gives a complete classification of the generalized Reed--Solomon locus into split semisimple, two nonsplit semisimple, and unipotent families. Over an algebraically closed field $k$, the Zariski closure $\GRSsurf_{r,k}$ of the semisimple GRS coefficient locus is an irreducible rational surface, generically parameterized two-to-one by a two-dimensional torus of geometric-progression root sets; reversal is the generic ambiguity. The affine quotient of the parameter torus by reversal is the normalization of $\GRSsurf_{r,k}\cap D(a_0)$, its nonzero-constant-term open part. The codimension in the space of monic degree-$r$ polynomials is $r-2$. Frobenius descent gives an exact formula for the number of GRS polynomials over $\mathbb F_q$. A canonical remainder parity-check matrix defines the MDS locus by a principal open condition. For fixed $r\ge3$ and $n\ge r+3$, the proportion of all monic degree-$r$ polynomials over $\mathbb F_q$ whose companion codes are MDS and non-GRS tends to one as $q\to\infty$ through prime powers.

math.NT↗

arXiv2026

On the inverses of permutation polynomials of the form $h(ψ(x))φ(x)+g(ψ(x))$ over finite fields

In this paper, we investigate the compositional inverses of permutation polynomials of the form \[ F(x)=h(ψ(x))φ(x)+g(ψ(x)) \in \mathbb{F}_{q^n}[x], \] where $ψ(x),φ(x) \in \mathbb{F}_{q^n}[x]$ are additive polynomials, $h(x), g(x) \in \mathbb{F}_{q^n}[x]$ satisfy $ h(ψ(\mathbb{F}_{q^n})) \subseteq \mathbb{F}_q^*, $ and there exists a polynomial $\barψ(x) \in \mathbb{F}_{q^n}[x]$ such that $ \barψ(F(x)) = ψ(x). $

math.NT↗

arXiv2026

The compositional inverses of the permutation polynomials of the form $x+γ\operatorname{Tr}_{q}^{q^n}(H(x))$ over $\mathbb{F}_{q^n}$

This paper focuses on computing the compositional inverses of permutation polynomials of the form $x+γ\operatorname{Tr}_{q}^{q^{n}}(H(x))$ over finite fields via the local method. We explicitly construct compositional inverses for four families of permutation polynomials of this type over $\mathbb{F}_{q^2}$, another four families over $\mathbb{F}_{q^3}$, and one general family over $\mathbb{F}_{q^n}$. The closed-form inverse expressions derived in this work supplement the theory of trace permutation polynomials.

math.NT↗

arXiv2026

Cyclic Codes and Cyclically Covering Subspaces over Finite Fields

Let $q$ be a power of a prime $p$, and let $n$ be a positive integer. A subspace $U\subseteq \mathbb F_q^n$ is called cyclically covering if the union of all its cyclic shifts covers $\mathbb F_q^n$, and $h_q(n)$ denotes the maximum possible codimension of such a subspace. This paper studies cyclically covering subspaces via cyclic codes. We first prove that $h_q(n)=0$ if and only if every nonzero cyclic code in $\mathbb F_q^n$ contains a full-weight codeword. We also relate $h_q(n)$ to the maximum weights of cyclic codes. In particular, when $h_q(n)>0$, we obtain sharp bounds for the maximum weight of cyclic codes without full-weight codewords and provide explicit examples attaining these bounds. Moreover, we study the number of cyclic codes containing no full-weight codeword. We determine this number completely over $\mathbb F_2$, and give lower bounds over $\mathbb F_3$. From this, we prove that if $q\ge 3$ is an odd prime and $m\ge 4$ is an integer, then $h_q\left(\frac{q^m+1}{2}\right)>0$.

math.NT↗

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

On Trivial Cyclically Covering Subspaces of $\mathbb{F}_q^n$ in Non-Coprime Characteristic

A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically covering} if the whole space $\mathbb{F}_q^n$ is the union of the cyclic shifts of $U$. The case $\mathbb{F}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd(n, q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if $n = p^k m$ where $p = \operatorname{char}(\mathbb{F}_q)$ and $\gcd(m, p)=1$, then $h_q(p^k m) = 0$ if and only if $h_q(m) = 0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.

math.NT↗

arXiv2025

Permutation Polynomials of the form $L(X)+γTr_q^{q^3}(h(X))$ over finite fields with even characteristic

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form $L(X)+Tr_q^{q^3}(h(X))$ have been constructed. This paper further investigates permutation polynomials of such form over $\mathbb{F}_{q^3}$. Unlike previous studies, we transform the problem of constructing univariate permutation polynomials over finite fields into that of constructing corresponding multivariate permutations over $\mathbb{F}_{q}$-vector spaces. Through this approach, we completely characterize a class of permutation polynomials of the form $L(X)+γTr_q^{q^3}(c_1X+c_2X^2+c_3X^3+c_4X^{q+2})$ over $\mathbb{F}_{q^3}$, where $q=2^m$, $L(X)=X^q+aX$ and $a,c_1,c_2,c_3,c_4,γ\in\mathbb{F}_q$ with $a^2+a+1\neq0$. Furthermore, using a similar method, we generalize several results from a recent work by Jiang, Li and Qu (2026).

math.NT↗

arXiv2025

New permutation polynomials over $\mathbb{F}_{q^2}$

In this paper, we propose a new method to obtain new permutation polynomials over $\mathbb{F}_{q^2}$. Using this method, we extend many known permutation polynomials, which take the form $\sum_i(x^q-x+δ)^{s_i}+L(x)$, where $L(x)$ is a $q$-polynomial over $\mathbb{F}_q$ and $δ\in\mathbb{F}_{q^2}$. We also present an alternative approach for constructing permutation polynomials of the form $x+γTr_q^{q^d}(x^{q+1}+x^{2q+2})$ for the cases where $q=2^m$, $2\nmid d$ and $ Tr_q^{q^d}(x)=x+x^q+\dots+x^{q^{d-1}}$.

math.NT↗

arXiv2025

The order of appearance of the product of the first and second Lucas numbers

Let $a$ and $b$ be relatively prime integers. Then the first Lucas sequence $\left(U_n\right)_{n\geq0}$ and the second Lucas sequence $\left(V_n\right)_{n\geq0}$ are defined respectively by $U_{n+2}=aU_{n+1}+bU_{n},\, U_0=0,\,U_1=1$ and $V_{n+2}=aV_{n+1}+bV_{n},\, V_0=2,\,V_1=a$, where $n\geq0$. Let $m$ be an integer with $\gcd(m,\,b)=1$. Then the smallest positive integer $k$ satisfying $m\mid U_k$ is called the order of appearance of $m$ in the first Lucas sequence $(U_n)_{n\geq0}$, denoted by $τ(m)$, i.e., $τ(m):=\min\{k\geq1:m\mid U_k\}$. When $a>0$ and $Δ=a^2+4b>0$, we give explicit formulae for $τ(U_m V_n), τ(U_m U_n)$, $τ(V_m V_n)$ and $τ(U_nU_{n+p}U_{n+2p})$, thus generalizing the results of Irmak and Ray.

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↗

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 $\mathbb{F}_{q^2}$.

math.NT↗

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.

math.NT↗