An extension of the Cobham-Semënov Theorem

2000 ◽  
Vol 65 (1) ◽  
pp. 201-211 ◽  
Author(s):  
Alexis Bès
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Pisot Numbers ◽  
Numeration Systems

AbstractLet θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

scholarly journals Nonderogatory Unicyclic Digraphs

2007 ◽  
Vol 2007 ◽  
pp. 1-8 ◽  
Author(s):  
Diego Bravo ◽  
Juan Rada
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Complete Product

A digraph is nonderogatory if its characteristic polynomial and minimal polynomial are equal. We find a characterization of nonderogatory unicyclic digraphs in terms of Hamiltonicity conditions. An immediate consequence of this characterization ia that the complete product of difans and diwheels is nonderogatory.

scholarly journals SIMETRISASI BENTUK KANONIK JORDAN

2021 ◽  
Vol 15 (1) ◽  
pp. 015-028
Author(s):  
Darlena Darlena ◽  
Ari Suparwanto
Keyword(s):  
Scalar Field ◽  
Linear Operator ◽  
Characteristic Polynomial ◽  
Canonical Form ◽  
Minimal Polynomial ◽  
Splitting Field ◽  
Jordan Decomposition ◽  
Jordan Canonical Form ◽  
Canonical Matrix

If the characteristic polynomial of a linear operator  is completely factored in scalar field of  then Jordan canonical form  of  can be converted to its rational canonical form  of , and vice versa. If the characteristic polynomial of linear operator  is not completely factored in the scalar field of  ,then the rational canonical form  of  can still be obtained but not its Jordan canonical form matrix . In this case, the rational canonical form  of  can be converted to its Jordan canonical form by extending the scalar field of  to Splitting Field of minimal polynomial   of , thus forming the Jordan canonical form of  over Splitting Field of  . Conversely, converting the Jordan canonical form  of  over Splitting Field of  to its rational canonical form uses symmetrization on the Jordan decomposition basis of  so as to form a cyclic decomposition basis of  which is then used to form the rational canonical matrix of

ON THE MINIMAL POLYNOMIAL OF A MATRIX

2004 ◽  
Vol 15 (01) ◽  
pp. 89-105
Author(s):  
THANH MINH HOANG ◽  
THOMAS THIERAUF
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Constant Term ◽  
Open Problems ◽  
Matrix Rank ◽  
The Matrix ◽  
Exact Counting ◽  
Class C

We investigate the complexity of the degree and the constant term of the minimal polynomial of a matrix. We show that the degree of the minimal polynomial is computationally equivalent to the matrix rank. We compare the constant term of the minimal polynomial with the constant term of the characteristic polynomial. The latter is known to be computable in the logspace counting class GapL. We show that if this holds for the minimal polynomial as well, then the exact counting in logspace class C=L is closed under complement. Whether C=L is closed under complement is one of the main open problems in this area. As an application of our techniques we show that the problem of deciding whether a matrix is diagonalizable is complete for AC0(C=L), the AC0-closure ofC=L.

scholarly journals The rank of Pseudo walk matrices: controllable and recalcitrant pairs

2020 ◽  
Vol 3 (3) ◽  
pp. 41-52
Author(s):  
Alexander Farrugia ◽  
Keyword(s):  
Lower Bound ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Minimal Polynomial ◽  
Full Rank ◽  
Gram Matrix ◽  
Largest Eigenvalue ◽  
The Largest Eigenvalue

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 \(\mathbf{v}\).

Pisot Numbers from {0, 1}-Polynomials

2010 ◽  
Vol 53 (1) ◽  
pp. 140-152
Author(s):  
Keshav Mukunda
Keyword(s):  
Unit Disk ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Pisot Number ◽  
Open Unit ◽  
Closed Unit Disk ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Salem Number ◽  
Newman Polynomials

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

scholarly journals The Pisot conjecture for -substitutions

2016 ◽  
Vol 38 (2) ◽  
pp. 444-472 ◽  
Author(s):  
MARCY BARGE
Keyword(s):  
Dynamical System ◽  
Discrete Spectrum ◽  
Minimal Polynomial ◽  
Pisot Number ◽  
Companion Matrix ◽  
Pisot Numbers ◽  
Almost Everywhere ◽  
One To One

We prove the Pisot conjecture for$\unicode[STIX]{x1D6FD}$-substitutions: if$\unicode[STIX]{x1D6FD}$is a Pisot number, then the tiling dynamical system$(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$associated with the$\unicode[STIX]{x1D6FD}$-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.

scholarly journals DIOPHANTINE SETS OF POLYNOMIALS OVER ALGEBRAIC EXTENSIONS OF THE RATIONALS

2014 ◽  
Vol 79 (3) ◽  
pp. 733-747
Author(s):  
CLAUDIA DEGROOTE ◽  
JEROEN DEMEYER
Keyword(s):  
Minimal Polynomial ◽  
Finite Extension ◽  
Algebraic Extension ◽  
Recursively Enumerable

AbstractLet L be a recursive algebraic extension of ℚ. Assume that, given α ∈ L, we can compute the roots in L of its minimal polynomial over ℚ and we can determine which roots are Aut(L)-conjugate to α. We prove that there exists a pair of polynomials that characterizes the Aut(L)-conjugates of α, and that these polynomials can be effectively computed. Assume furthermore that L can be embedded in ℝ, or in a finite extension of ℚp (with p an odd prime). Then we show that subsets of L[X]k that are recursively enumerable for every recursive presentation of L[X], are diophantine over L[X].

scholarly journals Convergence of the spectral radius of a random matrix through its characteristic polynomial

2021 ◽  
Author(s):  
Charles Bordenave ◽  
Djalil Chafaï ◽  
David García-Zelada
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Random Matrix

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

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