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Hai-Liang Wu

Publications and source records attributed to Hai-Liang Wu.

At least 19 recordsLinked to original sources
arXiv2026

On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums

Inspired by Weil's classical result on the zeta function of a projective Fermat curve defined over a finite field, in this paper, we investigate some arithmetic properties of the cyclotomic matrix $$\left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1},$$ where $p\ge3$ is a prime, $1\le k<p-1$ is a divisor of $p-1$ with $p-1=kn$, $χ$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$ and $J_p(χ^{ki},χ^{kj})$ is the Jacobi sum. For example, let $ζ_p\in\mathbb{C}$ be a primitive $p$-th root of unity and $P_k(T)$ be the minimal polynomial of the algebraic integer $$θ_k=\sum_{x\in\mathbb{F}_p,x^k=1}ζ_p^x$$ over $\mathbb{Q}$. Then we prove that $$\det \left[J_p(χ^{ki},χ^{kj})\right]_{1\le i,j\le n-1}=(-1)^{\frac{(k+1)(n^2-n)}{2}}\cdot n^{n-2}\cdot x_p(k),$$ where $x_p(k)$ is the coefficient of $T$ in $P_k(T)$.

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arXiv2026

Cyclic permutations of large subsets with polynomial values in multiplicative subgroups of finite fields

Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power for every consecutive pair. By combining mixed character-sum estimates, Fourier analysis on $\mathbb{F}_p$, and spectral graph methods, we establish a threshold $c(p,k,f)$ such that every subset $A$ with $\#A\ge c(p,k,f)$ admits such a cyclic ordering. We also give lower and upper bounds for the optimal threshold.

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arXiv2026

On the Burgess bound and Pythagorean triples involving primitive roots

By combining the recent results of Pierce and Xu on the higher-dimensional Burgess bound with the classical one-dimensional Burgess bound, we prove that for any sufficiently large prime $p$, there exists a Pythagorean triple $(a_p,b_p,c_p)$ with $a_p,b_p,c_p\in\mathbb{Z}\cap(0,p)$ such that $a_pb_p/2$ is a primitive root modulo $p$. This confirms a conjecture of Z.-W. Sun for all sufficiently large primes.

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arXiv2026

On Diophantine $m$-tuples related to primitive elements of finite fields

Inspired by recent works on Diophantine tuples over finite fields, in this paper we consider Diophantine tuples related to primitive elements of finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements and $\mathbb{F}_q^*=\mathbb{F}_q\setminus\{0\}$ be the multiplicative cyclic group of all non-zero elements over $\mathbb{F}_q$. An element $g\in\mathbb{F}_q$ is called primitive if $g$ generates the group $\mathbb{F}_q^*$. A set $\{x_1,x_2,\cdots,x_m\}\subseteq\mathbb{F}_q^*$ of $m$ elements is said to be a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ if $x_ix_j+1$ is primitive for any $1\le i\le j\le m$. Let $N_m$ denote the number of $\mathcal{P}$-Diophantine tuples over $\mathbb{F}_q$. Then we obtain the asymptotic formula $$m!\cdot N_m=\left(\frac{φ(q-1)}{q-1}\right)^{m(m+1)/2}q^m+O_{m,r}\left(q^{m-\frac{1}{2}+r}\right),$$ where $φ(\cdot)$ is the Euler totient function and $r\in(0, 1/2)$ is an arbitrary real number. Moreover, we prove that there exists a $\mathcal{P}$-Diophantine $m$-tuple over $\mathbb{F}_q$ whenever $q\ge \exp(\exp(m(m+1)))$.

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arXiv2026

The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix $$A_k(χ)=\left[G_N(χ^{ki+ki})\right]_{0\le i,j\le φ(N)/k-1},$$ where $N=p^m$ is a prime power, $φ(\cdot)$ is the Euler totient function, $k$ is a divisor of $φ(N)$, $χ$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and $$G_N(χ^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}χ^{ki+kj}(x)e^{2πix/N}$$ is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.

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arXiv2026

Cyclotomic matrices related to Kloosterman sums over finite fields

In this paper, by using the arithmetic properties of character sums over finite fields, we investigate some cyclotomic matrices involving Kloosterman sums over finite fields. For example, let $$K_q(u)=\sum_{x\in\mathbb{F}_q\setminus\{0\}}e^{\frac{2πi}{p}{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}\left(x+\frac{u}{x}\right)}$$ be the Kloosterman sum over $\mathbb{F}_q$, where $q=p^f$ is an odd prime power. We prove that matrix $[K_q(s_i+s_j)]_{1\le i,j\le (q-1)/2}$ is singular whenever $q\ge 11$, where $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.

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arXiv2026

Products involving the real parts of Jacobi sums and related cyclotomic matrices

Let $q$ be an odd prime power and $χ_q$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. In this paper, we study the arithmetic properties of the product $$R_q(χ_q)=\prod_{0<k<(q-1)/4}\left(J_q(ϕ_q,χ_q^k)+J_q(ϕ_q,χ_q^{-k})\right),$$ which is related to the real parts of Jacobi sums. Also, we reveal the connection between $R_q$ and the cyclotomic matrix $$\left[ϕ_q(s_i+s_j)\right]_{1\le i,j\le (q-1)/2},$$ where $ϕ_q$ is the unique quadratic multiplicative character of $\mathbb{F}_q$, and $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.

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arXiv2026

On new identities of Jacobi sums and related cyclotomic matrices

In this paper, using some arithmetic properties of Jacobi sums, we investigate some products involving Jacobi sums and reveal the connections between these products and certain cyclotomic matrices. In particular, as an application of our main results, we confirm a conjecture posed by Z.-W. Sun in 2019, and obtain a stronger result.

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arXiv2025

On restricted sums of four squares and Zhi-Wei Sun's $x+24y$ conjecture

In this paper, by using the arithmetic theory of ternary quadratic forms, we study some refinements on Lagrange's four-square theorem. For example, given positive integers $a,b$ satisfying some algebraic conditions and a positive integer $C\ge3$, we will show that for any sufficiently large integer $n$ with $\ord_2(n)\le C$, there exist non-negative integers $x,y,z,w$ such that $$ \begin{cases} x^2+y^2+z^2+w^2=n, ax+by\in\mathcal{S}, \end{cases} $$ where $\mathcal{S}$ is the set of all squares over $\mathbb{Z}$. In particular, we obtain some progress on Zhi-Wei Sun's $x+24y$ conjecture.

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arXiv2025

The Pell sequence and cyclotomic matrices involving squares over finite fields

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)/2)=0\Leftrightarrow Q_p\equiv 2\pmod{p^2\mathbb{Z}},$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i\}_{i=0}^{\infty}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.

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arXiv2025

Gaussian hypergeometric functions and cyclotomic matrices

Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all multiplicative characters of $\mathbb{F}_q$ and let $χ$ be a generator of $\widehat{\mathbb{F}_q^{\times}}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over $\mathbb{F}_q$. For example, let $s_1,s_2,\cdots,s_{(q-1)/2}$ be all nonzero squares over $\mathbb{F}_q$. For any integer $1\le r\le q-2$, define the matrix $$B_{q,2}(χ^r):=\left[χ^r(s_i+s_j)+χ^r(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}.$$ We prove that if $q\equiv 3\pmod 4$, then $$\det (B_{q,2}(χ^r))=\prod_{0\le k\le (q-3)/2}J_q(χ^r,χ^{2k})= \begin{cases} (-1)^{\frac{q-3}{4}}{\bf i}^nG_q(χ^r)^{\frac{q-1}{2}}/\sqrt{q} & \mbox{if}\ r\equiv 1\pmod 2,\\ G_q(χ^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2, \end{cases}$$ where $J_q(χ^r,χ^{2k})$ and $G_q(χ^r)$ are the Jacobi sum and the Gauss sum over $\mathbb{F}_q$ respectively.

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arXiv2025

On finite field analogues of determinants involving the Beta function

Motivated by the works of L. Carlitz, R. Chapman and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices concerning the Jacobi sums over finite fields, which can be viewed as finite field analogues of certain matrices involving the Beta function. For example, let $q>1$ be a prime power and let $χ$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. Then we prove that $$\det\left[J_q(χ^i,χ^j)\right]_{1\le i,j\le q-2}=(q-1)^{q-3},$$ where $J_q(χ^i,χ^j)$ is the Jacobi sum over $\mathbb{F}_q$. This is a finite analogue of $$\det [B(i,j)]_{1\le i,j\le n}=(-1)^{\frac{n(n-1)}{2}}\prod_{r=0}^{n-1}\frac{(r!)^3}{(n+r)!},$$ where $B$ is the Beta function. Also, if $q=p\ge5$ is an odd prime, then we show that $$\det \left[J_p(χ^{2i},χ^{2j})\right]_{1\le i,j\le (p-3)/2}=\frac{1+(-1)^{\frac{p+1}{2}}p}{4}\left(\frac{p-1}{2}\right)^{\frac{p-5}{2}}.$$

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arXiv2025

The Gross-Koblitz formula and almost circulant matrices related to Jacobi sums

In this paper, we mainly consider arithmetic properties of the cyclotomic matrix $B_p(k)=\left[J_p(χ^{ki},χ^{kj})^{-1}\right]_{1\le i,j\le (p-1-k)/k}$, where $p$ is an odd prime, $1\le k<p-1$ is a divisor of $p-1$, $χ$ is a generator of the group of all multiplicative characters of the finite field $\mathbb{F}_p$ and $J_p(χ^{ki},χ^{kj})$ is Jacobi sum over $\mathbb{F}_p$. By using the Gross-Koblitz formula and some $p$-adic tools, we first prove that $$p^{n-2}\det B_p(k)\equiv (-1)^{\frac{(n-1)(p+n-3)}{2}} \left(\frac{1}{k!}\right)^{n-2}\frac{1}{(2k)!}\pmod {p},$$ where $p-1=kn$. By establishing some theories on almost circulant matrices, we show that $$\det B_p(k)=(-1)^{\frac{(n-1)(p+n-1)}{2}}p^{-(n-1)}n^{n-2}a_p(k).$$ Here $a_p(k)$ is the coefficient of $t$ in the minimal polynomial of $\sum_{y\in U_k}(e^{2π{\bf i}y/p}-1)$, where $U_k$ is the set of all $k$-th roots of unity over $\mathbb{F}_p$. Also, for $k=1,2$ we obtain explicit expressions of $\det B_p(k)$.

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arXiv2024

A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields

Motivated by the recent work of Zhi-Wei Sun on determinants involving the Legendre symbol, in this paper, we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $ϕ$ be the unique quadratic multiplicative character of the finite field $\mathbb{F}_q$. If set $\{s_1,\cdots,s_{(q-1)/2}\}=\{x^2:\ x\in\mathbb{F}_q\setminus\{0\}\}$, then we prove that $$\det\left[t+ϕ(s_i+s_j)+ϕ(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}=\left(\frac{q-1}{2}t-1\right)q^{\frac{q-3}{4}}.$$ This confirms a conjecture of Zhi-Wei Sun.

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arXiv2024

On a generalization of R. Chapman's "evil determinant"

Let $p$ be an odd prime and $x$ be an indeterminate. Recently, Z.-W. Sun proposed the following conjecture: $$\det\left[x+\left(\frac{j-i}{p}\right)\right]_{0\le i,j\le \frac{p-1}{2}}=\begin{cases} (\frac{2}{p})pb_px-a_p & \mbox{if}\ p\equiv 1\pmod4, 1 & \mbox{if}\ p\equiv 3\pmod4, \end{cases}$$ where $a_p$ and $b_p$ are rational numbers related to the fundamental unit and class number of the real quadratic field $\mathbb{Q}(\sqrt{p})$. In this paper, we confirm the above conjecture of Sun based on Vsemirnov's decomposition of Chapman's "evil determinant".

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arXiv2024

On cyclotomic matrices involving Gauss sums over finite fields

Inspired by the works of L. Carlitz and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let $q=p^n$ be an odd prime power with $p$ prime and $n\in\mathbb{Z}^+$. Let $ζ_p=e^{2π{\bf i}/p}$ and let $χ$ be a generator of the group of all mutiplicative characters of the finite field $\mathbb{F}_q$. For the Gauss sum $$G_q(χ^{r})=\sum_{x\in\mathbb{F}_q}χ^{r}(x)ζ_p^{{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(x)},$$ we prove that $$\det \left[G_q(χ^{2i+2j})\right]_{0\le i,j\le (q-3)/2}=(-1)^{α_p}\left(\frac{q-1}{2}\right)^{\frac{q-1}{2}}2^{\frac{p^{n-1}-1}{2}},$$ where $$α_p= \begin{cases} 1 & \mbox{if}\ n\equiv 1\pmod 2, (p^2+7)/8 & \mbox{if}\ n\equiv 0\pmod 2. \end{cases}$$

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