We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor is no larger than the $n$-th Bell number, which is much smaller than the previously best-known upper bounds when $n \geq 4$. Over fields of non-zero characteristic we obtain even tighter upper bounds, and we also slightly improve the known lower bounds. In particular, we show that the $4 \times 4$ determinant over ${\mathbb{F}}_2$ has tensor rank exactly equal to $12$. Our results also improve upon the best-known upper bound for the Waring rank of the determinant when $n \geq 17$, and lead to a new family of axis-aligned polytopes that tile ${\mathbb{R}}^n$.

We determine the characteristic polynomials of the matrices $[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and $[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number $q\not =0,1$. As an application, for complex numbers $a,b,c$ with $b\not =0$ and $a^2\not =4b$, and the sequence $(w_m)_{m\in \mathbb Z}$ with $w_{m+1}=aw_m-bw_{m-1}$ for all $m\in \mathbb Z$, we determine the exact value of $\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.

We obtain asymptotic formulas for the number of matrices in the congruence subgroup

$$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$

Lehmer [‘On certain character matrices’, Pacific J. Math. 6 (1956), 491–499, and ‘Power character matrices’, Pacific J. Math. 10 (1960), 895–907] defines four classes of matrices constructed from roots of unity for which the characteristic polynomials and the kth powers can be determined explicitly. We study a class of matrices which arise naturally in transformation formulae of finite field hypergeometric functions and whose entries are roots of unity and zeroes. We determine the characteristic polynomial, eigenvalues, eigenvectors and kth powers of these matrices. The eigenvalues are natural families of products of Jacobi sums.

Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith. 115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting $a_1,\ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field $\mathbb {F}_q$ containing q elements with $2\nmid q$, we give the explicit value of the determinant $D_{(q-1)/2}=\det [(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if $q=p$ is a prime greater than $3$, then

$$ \begin{align*}\bigg(\frac{\det D_{(p-1)/2}}{p}\bigg)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4,\\ (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}\end{align*} $$

where $(\frac {\cdot }{p})$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb {Q}(\sqrt {-p})$.

For a principal ideal domain $A$, the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\textrm{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$. We prove that when $A[x]/(f(x))$ is maximal (i.e. integrally closed, i.e. a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

Given a set X of $n\times n$ matrices and a positive integer m, we consider the problem of estimating the cardinalities of the product sets $A_1 \cdots A_m$, where $A_i\in X$. When $X={\mathcal M}_n(\mathbb {Z};H)$, the set of $n\times n$ matrices with integer elements of size at most H, we give several bounds on the cardinalities of the product sets and solution sets of related equations such as $A_1 \cdots A_m=C$ and $A_1 \cdots A_m=B_1 \cdots B_m$. We also consider the case where X is the subset of matrices in ${\mathcal M}_n(\mathbb {F})$, where $\mathbb {F}$ is a field with bounded rank $k\leq n$. In this case, we completely classify the related product set.

As an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between left and right eigenvectors are derived. Two determinants of the matrices constructed by the left and right eigenvectors are evaluated in closed form.

Let $a,b$ and n be positive integers and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. For ${x\in S}$ , define $G_{S}(x)=\{d\in S: d<x, \,d\mid x \ \mathrm {and} \ (d\mid y\mid x, y\in S)\Rightarrow y\in \{d,x\}\}$ . Denote by $[S^a]$ the $n\times n$ matrix having the ath power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$ -entry. We show that the bth power matrix $[S^b]$ is divisible by the ath power matrix $[S^a]$ if $a\mid b$ and S is gcd closed (that is, $\gcd (x_i, x_j)\in S$ for all integers i and j with $1\le i, j\le n$ ) and $\max _{x\in S} \{|G_S (x)|\}=1$ . This confirms a conjecture of Shaofang Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008].

We use circulant matrices and hyperelliptic curves over finite fields to study some arithmetic properties of certain determinants involving Legendre symbols and kth power residues.

By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct. 30(7) (2019), 581–593].

In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is,

$$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$

$\chi $

$GL(n)$

${\mathbb {Q}}$

Given $d\in \mathbb{N}$, we establish sum-product estimates for finite, nonempty subsets of $\mathbb{R}^{d}$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, nonempty set of $d\times d$ diagonal matrices with real entries. Then, for all $\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$,

$$\begin{eqnarray}|A+A|+|A\cdot A|\gg _{d}|A|^{1+\unicode[STIX]{x1D6FF}_{1}/d},\end{eqnarray}$$

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$,

$$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$

$n\ges 222$

$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

Let $K/\mathbb{Q}$ be Galois and let $\eta \,\in \,{{K}^{\times }}$ be such that $\text{Re}{{\text{g}}_{\infty }}\left( \eta \right)\,\ne \,0$. We define the local $\theta $-regulator $\Delta _{p}^{\theta }\left( \eta \right)\,\in \,{{\mathbb{F}}_{p}}$ for the ${{\mathbb{Q}}_{p}}$-irreducible characters $\theta $ of $G=\,\text{Gal}\left( K/\mathbb{Q} \right)$. Let ${{V}_{\theta }}$ be the $\theta $-irreducible representation. A linear representation ${{\mathfrak{L}}^{\theta }}\,\simeq \,\delta \,{{V}_{\theta }}$ is associated with $\Delta _{p}^{\theta }\left( \eta \right)$ whose nullity is equivalent to $\delta \,\ge \,1$. Each $\Delta _{p}^{\theta }\left( \eta \right)$ yields $\text{R}eg_{p}^{\theta }\left( \eta \right)$ modulo $p$ in the factorization ${{\Pi }_{\theta }}{{\left( \text{Reg}_{p}^{\theta }\left( \eta \right) \right)}^{\phi \left( 1 \right)}}$ of $\text{Reg}_{p}^{G}\,\left( \eta \right)\,:=\frac{\text{Re}{{\text{g}}_{p}}\left( \eta \right)}{_{p}[K\,:\,\mathbb{Q}]}$ (normalized $p$-adic regulator). From Prob $\left( \Delta _{p}^{\theta }\left( \eta \right)=0\,\text{and}\,{{\mathfrak{L}}^{\theta }}\simeq \delta {{V}_{\theta }} \right)\,\le {{p}^{-f{{\delta }^{2}}}}$ ($f\,\ge \,1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that for $p$ large enough, $\text{Reg}_{p}^{G}\left( \eta \right)$ is a $p$-adic unit or ${{p}^{\phi \left( 1 \right)}}\,||\,\text{Reg}_{p}^{G}\left( \eta \right)$ (a single $\theta $ with $f\,=\,\delta \,=\,1$); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups ${{C}_{3,}}\,{{C}_{5}},\,{{D}_{6}}$) is conjecture would imply that for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational.We recall some deep cohomological results that may strengthen such conjectures.

We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles.

We find a lower bound on the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.

Let $\{x_i\}_{i=1}^{\infty}$ be an arbitrary strictly increasing infinite sequence of positive integers. For an integer $n\ge 1$, let $S_n=\{x_1,\ldots,x_n\}$. Let $\varepsilon$ be a real number and $q\ge 1$ a given integer. Let \smash{$\lambda _n^{(1)}\le \cdots\le \lambda _n^{(n)}$} be the eigenvalues of the power GCD matrix $((x_i, x_j)^{\varepsilon})$ having the power $(x_i,x_j)^{\varepsilon}$ of the greatest common divisor of $x_i$ and $x_j$ as its $i,j$-entry. We give a nontrivial lower bound depending on $x_1$ and $n$ for \smash{$\lambda _n^{(1)}$} if $\varepsilon>0$. Especially for $\varepsilon>1$, this lower bound is given by using the Riemann zeta function. Let $x\ge 1$ be an integer. For a sequence \smash{$\{x_i\}_{i=1}^{\infty }$} satisfying that $(x_i, x_j)=x$ for any $i\ne j$ and \smash{$\sum_{i=1}^{\infty }{1\over {x_i}}=\infty$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(1)}=x_1^{\varepsilon}-x^{\varepsilon }$}. Let $a\ge 0, b\ge 1$ and $e\ge 0$ be any given integers. For the arithmetic progression \smash{$\{x_{i-e+1}=a+bi\}_{i=e}^{\infty}$}, we show that if $0<\varepsilon\le 1$, then \smash{${\rm lim}_{n\rightarrow \infty }\lambda _n^{(q)}=0$}. Finally, we show that for any sequence \smash{$\{x_i\}_{i=1}^{\infty}$} and any \smash{$\varepsilon>0$, $\lambda_n^{(n-q+1)}$} approaches infinity when $n$ goes to infinity.