Irreducible Characteristic Polynomial Research Articles

Null ideals of sets of 3 × 3 similar matrices with irreducible characteristic polynomial

Let F be a field and the ring of matrices over F. Given a subset S of , the null ideal of S is the set of all polynomials f with coefficients from such that for all . We say that S is core if the null ideal of S is a two-sided ideal of the polynomial ring . We study sufficient conditions under which S is core in the case where S consists of matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if F is finite with q elements and , then S is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

Abstract 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 The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

Note on graphs with irreducible characteristic polynomials

Let G be a connected simple graph with characteristic polynomial PG(x). The irreducibility of PG(x) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P2n+1−2(n≥1) to certain vertices; union and join K1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results.

The rank of Pseudo walk matrices: controllable and recalcitrant pairs

A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) of a graph \(G\) having adjacency matrix \(\mathbf{A}\) is an \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\mathbf{A}\). We prove that the rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\mathbf{A}\) and a walk vector \(\mathbf{v}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable if \(\mathbf{W}_\mathbf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\mathbf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\) is a lower bound for the rank of \(\mathbf{W}_\mathbf{v}\). If the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound, then \((\mathbf{A},\mathbf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\mathbf{A},\mathbf{v})\) is neither controllable nor recalcitrant for any walk vector \(\mathbf{v}\).

Block conjugacy of irreducible toral automorphisms

ABSTRACTWe establish a matrix characterization (called block conjugacy) of the notion of weak equivalence of ideals. This gives the existence of a one-to-one correspondence between equivalence classes of block conjugate toral automorphisms (of a given irreducible characteristic polynomial) and equivalence classes of the weakly equivalent associated ideals. We show that there exist nonconjugate irreducible hyperbolic toral automorphisms A and B for which and are conjugate.

De Bruijn Sequences, Adjacency Graphs, and Cyclotomy

We study the problem of constructing De Bruijn sequences by joining cycles of linear feedback shift registers (LFSRs) with reducible characteristic polynomials. The main difficulty for joining cycles is to find the location of conjugate pairs between cycles, and the distribution of conjugate pairs in cycles is defined to be adjacency graphs. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)$ </tex-math></inline-formula> be a characteristic polynomial, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)=l_{1}(x)l_{2}(x)\cdots l_{r}(x)$ </tex-math></inline-formula> be a decomposition of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l(x)$ </tex-math></inline-formula> into pairwise co-prime factors. First, we show a connection between the adjacency graph of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l(x))$ </tex-math></inline-formula> and the association graphs of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l_{i}(x))$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\leq i\leq r$ </tex-math></inline-formula> . By this connection, the problem of determining the adjacency graph of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l(x))$ </tex-math></inline-formula> is decomposed to the problem of determining the association graphs of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {FSR}(l_{i}(x))$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1\leq i\leq r$ </tex-math></inline-formula> , which is much easier to handle. Then, we study the association graphs of LFSRs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields. At last, as an application of these results, we explicitly determine the adjacency graphs of some LFSRs and show that our results cover the previous ones.

On periodic properties of polylinear shift registers

Abstract This paper continues the author’s studies in [4], [5], [6] on synthesis of generators of pseudorandom sequences on the basis of polylinear shift registers. The paper [9] puts forward a method of the study of periodic properties of such automata. In the present paper, this method is used for the study of the cyclic type of a polylinear shift register with irreducible characteristic polynomials.

Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.

The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on the irreducible characteristic polynomial of degree $d$ and order $m$. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when $(q^{d}-1)/ m$ follows the quadratic residue conditions and also when it has the form $q^{2a}-q^{a}+1$ where $a\in \mathbb{N}$.

Counting integral points in certain homogeneous spaces

The leading term of asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the Brauer–Manin obstruction. The similar result is also true for homogeneous spaces of reductive groups with some restriction. As application, we will give the explicit asymptotic formulae of the number of integral points of certain norm equations and prove the leading term of asymptotic formula of the number of integral matrices with a fixed irreducible characteristic polynomial over Z studied by Eskin–Mozes–Shah is equal to the product of the number of local integral solutions over all primes although the density function defined by Borovoi and Rudnick is not trivial in general. We also answer a question raised by Borovoi and Rudnick for comparing the number of integral symmetric matrices with the given determinant with the product of local densities.

The Structure of Polynomial Invariants of Linear Loops

. This article considers the problem of generating polynomial invariants for iterative loops with loop initialization statements and nonsingular linear operators in loop bodies. The set of such invariants forms an ideal in the ring of polynomials in the loop variables. Two algorithms are presented one of which calculates basic invariants for a linear operator in the form of a Jordan cell and the other calculates basic invariants for a diagonalizable linear operator with an irreducible minimal characteristic polynomial. The following theorem on the structure of the basis of the ideal of invariants for such an operator is proved: this basis consists of basic invariants of Jordan cells and basic invariants of the diagonalizable part of the linear operator being considered.

Structure of the sets of mutually unbiased bases with cyclic symmetry

Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, for they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field 2 equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.

Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture

We relate the construction of a complete set of cyclic mutually unbiased bases, i.e., mutually unbiased bases generated by a single unitary operator, in power-of-two dimensions to the problem of finding a symmetric matrix over \documentclass[12pt]{minimal}\begin{document}$\mathbb {F}_2$\end{document}F2 with an irreducible characteristic polynomial that has a given Fibonacci index. For dimensions of the form \documentclass[12pt]{minimal}\begin{document}$2^{2^k}$\end{document}22k, we present a solution that shows an analogy to an open conjecture of Wiedemann in finite field theory. Finally, we discuss the equivalence of mutually unbiased bases.

Conjugacy Classes and Class Number

AbstractIt is shown that the conjugacy classes of integral matrices with a given irreducible characteristic polynomial is in bijection with the class group of a corresponding order in an algebraic number field.

Walks on graphs and lattices – effective bounds and applications

We continue the investigations started in [Rivin, Technical Report math., 1999, Rivin, Duke Math. J., 2006]. We consider the following situation: G is a finite directed graph, where to each vertex of G is assigned an element of a finite group Γ. We consider all walks of length N on G , starting from ν i and ending at ν j . To each such walk w we assign the element of Γ equal to the product of the elements along the walk. The set of all walks of length N from ν i to ν j thus induces a probability distribution F N,i,j on Γ. In [Rivin, Technical Report math., 1999] we give necessary and sufficient conditions for the limit as N goes to infinity of F N,i,j to exist and to be the uniform density on Γ (a detailed argument is presented in [Rivin, Duke Math. J., 2006]). The convergence speed is then exponential in N . In this paper we consider ( G , Γ), where Γ is a group possessing Kazhdan's property T (or, less restrictively, property τ with respect to representations with finite image), and a family of homomorphisms ψ k : Γ → Γ k with finite image. Each F N,i,j induces a distribution on Γ k (by push-forward under ψ k ). Our main result is that, under mild technical assumptions, the exponential rate of convergence of to the uniform distribution on Γ k does not depend on k . As an application, we prove effective versions of the results of [Rivin, Duke Math. J., 2006] on the probability that a random (in a suitable sense) element of SL( n , ℤ) or Sp( n , ℤ) has irreducible characteristic polynomial, generic Galois group, etc.

A uniqueness theorem on matrices and reconstruction

We prove the following simple uniqueness theorem: Let A and B be two integral symmetric matrices with the same irreducible characteristic polynomial. If there exist rational orthogonal matrices Q such that B = Q T A Q , then Q is unique up to a sign. An application of the theorem to the reconstruction conjecture is provided.

Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms

We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically, SL(n,Z) and Sp(2n,Z)) to show that a random element in one of these lattices has irreducible characteristic polynomials (over Z). The term random can be defined in at least two ways: first, in terms of height; second, in terms of word length in terms of a generating set. We show the result using both definitions. We use these results to show that a random (in terms of word length) element of the mapping class group of a surface is pseudo-Anosov and that a random free group automorphism is irreducible with irreducible powers (or fully irreducible*)

Maximum sequence test pattern generators with irreducible characteristic polynomials

Abstract Mechanisms that can generate maximum-length sequences (M-sequences) are of particular interest in pseudoexhaustive or pseudorandom built-in Test Pattern Generation (TPG). The characteristic polynomial of these mechanisms is restricted to be chosen among primitive polynomials only, since the latter require only one initialization state (seed). Recently, it was shown that any non-primitive irreducible polynomial can be used as characteristic polynomial to generate an M-sequence in the minimum number of cycles and with hardware overhead bounded by a low constant irrespectively of the number of seeds. In this paper, we describe and analyze the hardware overhead of two alternative schemes for the same purpose. These schemes offer even lower hardware overhead than the original one for a large majority of non-primitive irreducible polynomials of each degree. We also catalog each non-primitive irreducible polynomial of degree 12–22 according to the least costly of these schemes that can be used with it to generate an M-sequence.

Signature Analysis for Test Responses of Sequential Circuits

Many test schemes use signature analyzers to compact the responses of a circuit under test. Unfortunately, there can be some faulty circuits with erroneous test responses but exactly the same signature as in the fault-free case. Hence, methods are required to determine how many faults become undetectable due to aliasing. Whereas previous work concentrated on combinational circuits, this paper investigates signature analysis for a wide range of sequential circuits, where the errors in successive responses are correlated. It is shown that for almost all faults of these circuits the probability of aliasing in a signature analyzer with k bits asymptotically approaches 2−k or is 0 if a signature analyzer with an irreducible characteristic polynomial is used and certain test lengths are avoided. The limiting value can be used as a good approximation for practical test lengths. These results are particularly useful for advanced built-in self-test techniques with low hardware overhead.

A galois approach to mth roots of matrices with rational entries

Let A be a given n × n matrix with rational entries and irreducible characteristic polynomial f( x). We investigate the Galois groups of f( x) and f( x m ), to find necessary and sufficient conditions for the existence of a solution B to the matrix equation A = B m , where B is also a matrix with rational entries. We do this by finding necessary and sufficient conditions for f( x m ) to have a factor of degree n (with rational coefficients).