Approximations of algebraic irrationalities with matrices

Euler matrices and their algebraic properties revisited

This paper is concerned with the generalized Euler polynomial matrix $\E^{(α)}(x)$ and the Euler matrix $\E$. Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for $\E^{(α)}(x)$ and determine the inverse matrix of $\E$. We establish some explicit expressions for the Euler polynomial matrix $\E(x)$, which involving the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in …

Arithmetic matrices for number fields I

We provide a simple way to add, multiply, invert, and take traces and norms of algebraic integers of a number field using integral matrices. With formulas for the integral bases of the ring of integers of at least a significant proportion of numbers fields, we obtain explicit formulas for these matrices and discuss their generalization. These results are useful in proving statements about the particular number fields they work in. We give a meaningful diagonalization that is helpful in this regard and can be used to define such matrices in general. The matrix identities provided suggest that c…

Conjugacy of Integral Matrices over Algebraic Extensions

We consider conjugacy of integral matrices by elements in $\text{GL}_{n}(R)$ for certain rings $R$ with subring $\mathbb{Z}$. We note that a Hasse principal does not hold in the context of matrix conjugacy because matrices which are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all $p$ are not necessarily $\text{GL}_{n}(\mathbb{Z})$-conjugate. By a theorem of Guralnick, we know that integral $n \times n$ matrices are $\text{GL}_{n}(\mathbb{Z}_p)$-conjugate for all primes $p$ if and only if they are conjugate by an element in $\text{GL}_{n}(E)$ for some algebraic integral extension $E$ of $\mathb…

Waring's problem for matrices over orders in algebraic number fields

In this paper we give necessary and sufficient trace conditions for an n by n matrix over any commutative and associative ring with unity to be a sum of k-th powers of matrices over that ring, where n,k are integers greater equal 2. We prove a discriminant criterion for every 2 by 2 matrix over an order R to be sums of cubes and fourth powers over R. We also show that if q is a prime and n greater equal 2, then every n by n matrix over the ring of integers O, of a quadratic number field is a sum of q-th powers (of matrices) over O if and only if q is coprime to the discriminant of the quadrati…

代数数的对数矩阵的秩 I：Baker与Waldschmidt-Masser定理

Ranks of matrices of logarithms of algebraic numbers I: the theorems of Baker and Waldschmidt-Masser

本文综述了代数数对数矩阵的秩，证明了Baker定理弱形式与Waldschmidt-Masser定理，并提出了关于方阵奇异性的矩阵系数新猜想。

通过二元3×3矩阵研究雅可比施塔尔数的矩阵表示与算术性质

Matrix representations and arithmetic properties of jacobsthal numbers via binary 3x3 matrices

该研究通过二元3×3矩阵的线性代数方法，推导雅可比施塔尔数的矩阵幂公式与多种恒等式，分析其算术性质，确定生成该序列的共轭类数量，建立矩阵理论与二阶线性递推的统一框架。

Parallel computation of the rank of large sparse matrices from algebraic K-theory

This paper deals with the computation of the rank and of some integer Smith forms of a series of sparse matrices arising in algebraic K-theory. The number of non zero entries in the considered matrices ranges from 8 to 37 millions. The largest rank computation took more than 35 days on 50 processors. We report on the actual algorithms we used to build the matrices, their link to the motivic cohomology and the linear algebra and parallelizations required to perform such huge computations. In particular, these results are part of the first computation of the cohomology of the linear group GL_7(Z…

组合矩阵的代数方面

Algebraic Aspects of combined matrices

本文研究了数域$K$上非奇异矩阵$A$的组合矩阵$\mathcal{C}(A)$的代数性质，分析了$A$属于$\mathrm{GL}_n(K)$代数子群的情况，给出了$n=2$和$n=3$时$\mathcal{C}(A)$的显式对角化结果。

Equidistribution of expanding translates of curves and Diophantine approximation on matrices

We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/Γ$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Dio…

通过矩阵研究Picard堆的群律

The group law of Picard stacks via matrices

本文研究了通过矩阵构建严格交换Picard堆的代数性，证明其等价于适合的代数2堆理论下的2代数堆。

On metric diophantine approximation in matrices and Lie groups

We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over the rationals. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.

Cohen Lenstra Partitions and Mutually Annihilating Matrices over a Finite Field

Motivated by questions in algebraic geometry, Yifeng Huang recently derived generating functions for counting mutually annihilating matrices and mutually annihilating nilpotent matrices over a finite field. We give a different derivation of his results using statistical properties of random partitions chosen from the Cohen-Lenstra measure.