A directed graph G is nonderogatory if its adjacency matrix A is nonderogatory, i.e., the characteristic polynomial of A is equal to the minimal polynomial of A. We analyze the problem whether the coalescence of difans and diwheels is nonderogatory. Also, a formula for the characteristic polynomial of the coalescence of two directed graphs is presented. 2000 Mathematics Subject Classification: ...

The notions separability and normality are related to this characterisation. In the case of skew fields polynomials often have infinitely many zeros, so a different way of counting zeros as distinct is needed. The well-known theorem of Gordon and Motzkin [Z] states that a polynomial of degree n has zeros in at most n conjugacy classes. This suggests one should count zeros of a polynomial by the...

The Cayley Hamilton theorem on the characteristic polynomial of a matrix A and Frobenius’ theorem on minimal polynomial of A are deduced from the familiar Laplace transform formula L(eAt) = (sI − A)−1. This formula is extended to a formal power series ring over an algebraically closed field of characteristic 0, so that the argument applies in the more general setting of matrices over a field of...

The group G = GL2(Fq) is not only the slightly overweight cousin of GL2(C) (it is a “Flabby” S2) but also a tremendously interesting finite group. It is both the automorphism group of the vector space of dimension 2 over the field Fq and a finite group of order q(q − 1)(q − 1), so naturally, representation theorists would like very much to understand its representations. We follow the expositio...

In this paper we show that the index of a 1-reducible subgroup of the diierential Galois group of an ordinary homogeneous linear diierential equation L(y) = 0 yields the best possible bound for the degree of the minimal polynomial of an algebraic solution of the Riccati equation associated to L(y) = 0. For an irreducible third order equation we show that this degree belongs to f3;6;9;21;36g. Wh...

It is unusual for an irreducible polynomial to have a root with rational real part or with rational imaginary part. Of course, such polynomials exist: one can simply take the minimal polynomial of, say, 1+ i √ 2 or √ 2+ i. The same applies to polynomials having a root of rational modulus. But it turns out to be of interest to characterize these three kinds of polynomials. We therefore define ou...

Given a non-hermitean matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note will explain how to determine the minimal polynomial of a matrix without going through its characteristic polynomial. The approach is applied to a quantum mechanical particle moving in a square well under the influence of a piece-wise constant PT-symmetric potential. Upo...

We describe an iterative method of constructing some favorable auxiliary polynomials used to obtain lower bounds in some problems of algebraic number theory. With this method we improve a lower bound on Mahler’s measure of a polynomial with no cyclotomic factors whose coefficients are all congruent to 1 modulo m for some integer m ≥ 2, raise a lower bound in the problem of Schinzel and Zassenha...

We propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix F (s): The proposed method exploits the structure of the left null space of generalized Wolovich or Sylvester resultants to compute row polynomial vectors that form a minimal polynomial basis of left kernel of the given polynomial matrix. The entire procedure can be imple...

In this article, we study the problem of minimizing resource capacity autonomous agents that cooperate to achieve a shared task. More specifically, consider high-level planning for team homogeneous operate under constraints in stochastic environments and share common goal: given set target locations, ensure each location is visited infinitely often by some almost surely. We formalize dynamics s...