Related papers: Representation of Cyclotomic Fields and Their Subf…

We present a new type of equivalence for representable matroids that uses the automorphisms of the underlying matroid. Two $r\times n$ matrices $A$ and $A'$ representing the same matroid $M$ over a field $F$ are {\it geometrically…

Given two cyclotomic polynomials $\Phi_n(x)$ and $\Phi_m(x)$, $n\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)\Phi_n (x)+b(x)\Phi_m(x),$$ with a(x) and b(x) integer polynomials.

Let $K[HK_{\Theta}]$ denote the Hecke-Kiselman algebra of a finite oriented graph $\Theta$ over an algebraically closed field $K$. All irreducible representations, and the corresponding maximal ideals of $K[HK_{\Theta}]$, are characterized…

Let $r \leqslant n$ be nonnegative integers, and let $N = \binom{n}{r} - 1$. For a matroid $M$ of rank $r$ on the finite set $E = [n]$ and a partial field $k$ in the sense of Semple--Whittle, it is known that the following are equivalent:…

For W_N minimal model CFT's at Large N, we formulate a nonlinear field theory of primary operators. A classification of single-trace operators is given first based on which an interacting field theory operating in Fock space is built. A…

It is proved that for each prime field $GF(p)$, there is an integer $f(p)$ such that a 4-connected matroid has at most $f(p)$ inequivalent representations over $GF(p)$. We also prove a stronger theorem that obtains the same conclusion for…

We extend the notion of representation of a matroid to algebraic structures that we call skew partial fields. Our definition of such representations extends Tutte's definition, using chain groups. We show how such representations behave…

For a central division algebra $D$ of dimension $d^2$ over a finite extension $F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristic prime to $p$, and an irreducible smooth $R$-representation $\pi$ of $G=GL_n(D)$, we…

Using the properties of the ideal of the coordinate Hermite interpolation on n-dimensional grid [4], we prove that the extension k in k[x1, x2, ..., xn] / (f1(x1), ..., fn(xn)) has a primitive element if and only if at most one of the…

It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…

For a square-free bivariate polynomial $p$ of degree $n$ we introduce a simple and fast numerical algorithm for the construction of $n\times n$ matrices $A$, $B$, and $C$ such that $\det(A+xB+yC)=p(x,y)$. This is the minimal size needed to…

The first problem we investigate is the following: given $k\in \mathbb{R}_{\ge 0}$ and a vector $v$ of Pl\"ucker coordinates of a point in the real Grassmannian, is the vector obtained by taking the $k$th power of each entry of $v$ again a…

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…

We develop a Galois descent approach to finite-field Fourier spectra over an arbitrary finite base field. Let $\mathbb K=\mathbb F_q$ and $\mathbb L=\mathbb F_{q^m}$. If a Fourier transform is applied to a $\mathbb K$-valued vector, then…

In this paper we provide an elementary and easy proof that a proper subalgebra of the matrix algebra $ \mathbb{K}^{n,n }$, with $n \geq3$ and $\mathbb{K}$ an arbitrary field, has dimension strictly less than $n^2-1$.

Let $\mathbb{F}$ be a field. We show that given any $n$th degree monic polynomial $q(x)\in \mathbb{F}[x]$ and any matrix $A\in\mathbb{M}_n(\mathbb{F})$ whose trace coincides with the trace of $q(x)$ and consisting in its main diagonal of…

A matroid is $\text{GF}(q)$-regular if it is representable over all proper superfields of the field $\text{GF}(q)$. We show that, for highly connected matroids having a large projective geometry over $\text{GF}(q)$ as a minor, the property…

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…

A matrix is said to be {\it cyclic} if its characteristic polynomial is equal to its minimal polynomial. Cyclic matrices play an important role in some algorithms for matrix group computation, such as the Cyclic Meataxe developed by P. M.…

Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.