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Hai-Liang Wu

Publications and source records attributed to Hai-Liang Wu.

At least 37 records · Page 2Linked to original sources
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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arXiv2024

On generalized Legendre matrices involving roots of unity over finite fields

In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting $a_1,\cdots,a_{(q-1)/2}$ be all non-zero squares in the finite field $\mathbb{F}_q$ which contains $q$ elements with $2\nmid q$, we give the explicit value of $D_{(q-1)/2}=\det[(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if $q=p$ is a prime greater than $3$, then $$\left(\frac{\det D_{(p-1)/2}}{p}\right)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4, (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}$$ where $(\cdot/p)$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$.

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arXiv2024

On sumsets involving $k$th powers of finite fields

In this paper, we study some topics concerning the additive decompositions of the set $D_k$ of all $k$th power residues modulo a prime $p$. For example, given a positive integer $k\ge2$, we prove that $$\lim_{x\rightarrow+\infty}\frac{B(x)}{π(x)}=0,$$ where $π(x)$ is the number of primes $p\le x$ and $B(x)$ denotes the cardinality of the set $$\{p\le x: p\equiv1\pmod k; D_k\ \text{has a non-trivial 2-additive decomposition}\}.$$

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arXiv2024

Arithmetic progressions represented by diagonal ternary quadratic forms

Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called $(d,r)$-universal. In this paper, via the theory of ternary quadratic forms we study the $(d,r)$-universality of some diagonal ternary quadratic forms conjectured by L. Pehlivan and K. S. Williams, and Z.-W. Sun. For example, we prove that $2x^2+3y^2+10z^2$ is $(8,5)$-universal, $x^2+3y^2+8z^2$ and $x^2+2y^2+12z^2$ are $(10,1)$-universal and $(10,9)$-universal, and $3x^2+5y^2+15z^2$ is $(15,8)$-universal.

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arXiv2022

Additive decompositions of cubes in finite fields

Let $p\equiv1\pmod3$ be a prime . We study several topics on additive decompositions concerning the set $C_p$ of all non-zero cubes in the finite field of $p$ elements. For example, we show that when $p>184291$ , the set $C_p$ has no decomposition of the form $C_p=A+B+C$ with $|A|,|B|,|C|\ge2$.

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arXiv2022

A conjecture of Zhi-Wei Sun on determinants over finite fields

In this paper, we study certain determinants over finite fields. Let $\mathbb{F}_q$ be the finite field of $q$ elements and let $a_1,a_2,\cdots,a_{q-1}$ be all nonzero elements of $\mathbb{F}_q$. Let $T_q=\left[\frac{1}{a_i^2-a_ia_j+a_j^2}\right]_{1\le i,j\le q-1}$ be a matrix over $\mathbb{F}_q$. We obtain the explicit value of $\det T_q$. Also, as a consequence of our result, we confirm a conjecture posed by Zhi-Wei Sun.

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arXiv2021

$q$-Supercongruences from transformation formulas

Let $Φ_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*} &\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\ &\quad\equiv\begin{cases}0 \pmod{[n]Φ_n(q)^2}, &\text{if }n\equiv 1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases} \end{align*} Applying the `creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the Chinese remainder theorem for coprime polynomials, we confirm the above conjecture, as well as another similar $q$-supercongruence conjectured by Guo and Schlosser.

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arXiv2021

Elliptic curves over $\mathbb{F}_p$ and determinants of Legendre matrices

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d\ne0$ and $c^2-4d\ne0$, we show that there are infinitely many odd primes $p$ such that $$\det\bigg[\left(\frac{i^2+cij+dj^2}{p}\right)\bigg]_{0\le i,j\le p-1}=0,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.

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arXiv2021

Cubes in finite fields and related permutations

Let $p=3n+1$ be a prime with $n\in\mathbb{N}=\{0,1,\cdots\}$, and let $g\in\mathbb{Z}$ be a primitive root modulo $p$. Let $0<a_1<\cdots<a_n<p$ be all the cubic residues modulo $p$ in the interval $(0,p)$. Then clearly the sequence $$a_1\ {\rm mod}\ p,\ a_2\ {\rm mod}\ p,\cdots, a_n\ {\rm mod}\ p$$ is a permutation $s_p(g)$ of the sequence $$g^3\ {\rm mod}\ p,\ g^6\ {\rm mod}\ p,\cdots, g^{3n}\ {\rm mod}\ p.$$ In this paper, we shall determine the sign of this permutation.

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arXiv2021

Primitive elements and $k$-th powers in finite fields

Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in $\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if $q>\max\{e^{e^3},(2k)^6\}$, then there is a primitive element $g$ of $\mathbb{F}_q$ such that $f(g)\in\mathbb{F}_q^{\times k}=\{x^k: x\in\mathbb{F}_q\setminus\{0\}\}$. Moreover, we shall confirm a conjecture posed by Sun.

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arXiv2021

Products of quadratic residues and related identities

In this paper we study products of quadratic residues modulo odd primes and prove some identities involving quadratic residues. For instance, let $p$ be an odd prime. We prove that if $p\equiv5\pmod8$, then $$\prod_{0<x<p/2,(\frac{x}{p})=1}x\equiv(-1)^{1+r}\pmod p,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol and $r$ is the number of $4$-th power residues modulo $p$ in the interval $(0,p/2)$. Our work involves class number formula, quartic Gauss sums, Stickelberger's congruence and values of Dirichlet L-series at negative integers.

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arXiv2021

Quadratic residues and related permutations

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For instance, we consider the following three sequences. \begin{align*} &A_0: \overline{1^2},\ \overline{2^2},\ \cdots,\ \overline{((p-1)/2)^2},\\ &A_1: \overline{a_1},\ \overline{a_2},\ \cdots,\ \overline{a_{(p-1)/2}},\\ &A_2: \overline{g^2},\ \overline{g^4},\ \cdots,\ \overline{g^{p-1}}, \end{align*} where $g\in\Z$ is a primitive root modulo $p$ and $1\le a_1<a_2<\cdots<a_{(p-1)/2}\le p-1$ are all quadratic residues modulo $p$. Obviously $A_i$ is a permutation of $A_j$ and we call this permutation $σ_{i,j}$. Sun obtained the sign of $σ_{0,1}$ when $p\equiv 3\pmod4$. In this paper we give the sign of $σ_{0,1}$ and determine the sign $σ_{0,2}$ when $p\equiv 1\pmod 4$.

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arXiv2021

Almost universal ternary sums of pentagonal numbers

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $aP_5(x)+2^rbP_5(y)+2^scP_5(z)$ represents all but finitely many positive integers.

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arXiv2021

Frobenius matrices and a variant of Zolotarev's theorem

In this paper, with the help of the theory of matrices and finite fields we generalize Zolotarev's theorem to an arbitrary finite dimensional vector space over $\mathbb{F}_q$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements.

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