Nguyen Van The
In this short note, we prove that for $\mathcal{E} \subset \mathbb{F}_q^d$ with $|\mathcal{E}| \geq q^{d-1} + O(q^2)$ then the set of determinants generated by $\mathcal{E}$ is $\mathbb{F}_q.$ This result is nearly optimal and generalizes the previous results of Vinh (2013) and Iosevich, Rudnev and Zhai (2015).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Paratrophic Determinants over $\mathbb{Z}/N\mathbb{Z}$ via Discrete Fourier Transform
本文通过离散傅里叶变换研究$\mathbb{Z}/N\mathbb{Z}$上的para-trophic行列式，证明其可分解为由N的因数d索引的群行列式乘积，并给出包含周期伯努利函数和正切函数幂的行列式族的显式公式，同时修正了Sun Zhi-Wei的猜想。
Let $\mathbb Z_n$ denote the cyclic group of order $n$. We show how the group determinant for $G= \mathbb Z_n \times H$ can be simply written in terms of the group determinant for $H$. We use this to get a complete description of the integer group determinants for $\mathbb Z_2 \times D_8$ where $D_8$ is the dihedral group of order 8, and $\mathbb Z_2 \times Q_8$ where $Q_8$ is the quaternion group of order 8.
Let $\mathbb{F}_q$ be an arbitrary finite field of order $q$. In this article, we study $\det S$ for certain types of subsets $S$ in the ring $M_2(\mathbb F_q)$ of $2\times 2$ matrices with entries in $\mathbb F_q$. For $i\in \mathbb{F}_q$, let $D_i$ be the subset of $M_2(\mathbb F_q)$ defined by $ D_i := \{x\in M_2(\mathbb F_q): \det(x)=i\}.$ Then our results can be stated as follows. First of all, we show that when $E$ and $F$ are subsets of $D_i$ and $D_j$ for some $i, j\in \mathbb{F}_q^*$, respectively, we have $$\det(E+F)=\mathbb F_q,$$ whenever $|E||F|\ge {15}^2q^4$, and then provide a c…
The integer group determinants for the abelian groups of order 18
本文完整描述了18阶两个阿贝尔群的整数群行列式，包括循环行列式情形，从而完成了对小于20阶群的研究。
In this note we prove an assertion made by M. Levin in 1999: the Pascal matrix modulo 2 has the property that each of the square sub-matrices laying on the upper border or on the left border has determinants, computed in $\mathbb{Z}$, equal to 1 or -1.
The integer group determinants for $\mathbb{Z}_p^n$
本文研究 $\mathbb{Z}_p^n$ 的整数群行列式，给出 $p$ 的允许幂次的新条件，证明 $n\leq 5$ 时精确，并完整刻画 $\mathbb{Z}_3^3$。
Counting integer matrices with square-free determinants
研究整数矩阵集合中行列式为平方自由数的矩阵数量，并推导出相关渐进行为，同时改进方法计算矩阵行列式与欧拉函数的和。
The integer group determinants for GA(1,p) and related semidirect products
本文研究 Z_p 与 Z_n 半直积群的整数群行列式，完整刻画若干 GA(1,p) 及相关群，并证明自然整除与同余条件在特定情形下既必要又充分。
Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.
A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of ${\mathbb Z}^2$, as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations…
We consider the values taken by $n\times n$ circulant determinants with integer entries when $n$ is the product of two distinct odd primes $p,q$. These correspond to the integer group determinants for $\mathbb Z_{pq}$, the cyclic group of order $pq$. We show that $p^2$ and $q^2$ are not determinants (more generally we show that the classic necessary divisibility conditions are never sufficient when $n$ contains at least two odd primes). We obtain a complete description of the integer group determinants for $\mathbb Z_{15}$ (the smallest unresolved group) and partial results for general $n=3p.$
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.
We consider the KZ differential equations over $\mathbb C$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $\mathbb F_p$. We study the polynomial solutions of these differential equations over $\mathbb F_p$, constructed in a previous work joint with V.\,Schechtman and called the $\mathbb F_p$-hypergeometric solutions. The dimension of the space of $\mathbb F_p$-hypergeometric solutions depends on the prime number $p$. We say that the KZ equations have ample reduction for a prime $p$, if the dimen…
The $5$-divisible integer group determinants for the elementary abelian group of order 25
本文针对25阶初等阿贝尔群C₅×C₅，证明其5-可整除群行列式的反向包含关系，结合已有结果完成25阶群的Taussky-Todd整数群行列式问题分类。
In previous papers we investigated basic properties of the determinant $G_{K}(s)$ of the Riemann operator: ${\mathcal R}$ acting on $\bigoplus_{n>1} K_{n}(A)_{\mathbb{C}}$, where $A$ is the integer ring of an algebraic number field $K$. The function $G_{K}(s)$ is defined as the regularized determinant \[ G_{K}(s) = {\rm det} ((s I-\mathcal{R}) | \bigoplus_{n>1} K_{n}(A)_{\mathbb{C}} ) \] with $\mathcal{R} | K_{n}(A)_{\mathbb{C}} = \frac{1-n}{2}$. We showed that $G_{K}(s)^{-1}$ is essentially the so called gamma factors of Dedekind zeta function of $K$. In this paper we study the transcendency …
Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$ D_n(A)=\{D\in \mathbb{F}\ |\ \exists a_{ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ij})\bigr)=D\}, $$ where the symbol $(a_{ij})$ defines the matrix with elements $a_{ij}$. How big is the size of the set $D_n(A)$ comparing to the size of the set $A$?
Elliptic curves over $\mathbb{F}_p$ and determinants of Legendre matrices
本文研究以 Legendre 符号为元素的行列式与有限域椭圆曲线的关系，证实了 Sun 提出的若干猜想，并证明了存在无穷多个奇素数使得特定矩阵的行列式为零。
We prove that $p$-determinants of a certain class of differential operators can be lifted to power series over $\mathbb{Q}$. We compute these power series in terms of monodromy of the corresponding differential operators.
A geometric determinant method and geometric dimension growth
本文研究几何维度增长猜想的几何版本，通过发展几何行列式方法，证明当X的度数d≥6时，X(b)的不可约分量维数为b dim X的个数受d的多项式限制。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。