FAST PARALLEL COMPUTATION OF THE JORDAN NORMAL FORM OF MATRICES

1996 ◽  
Vol 06 (02) ◽  
pp. 203-212 ◽  
Author(s):  
JEAN-LOUIS ROCH ◽  
GILLES VILLARD
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Parallel Computation ◽  
Characteristic Zero ◽  
Jordan Normal Form

We establish that the problem of computing the Jordan normal form of a matrix over a field F is in [Formula: see text] for F being a field of characteristic zero or a finite field.

Limit Jordan normal form of large triangular matrices over a finite field

1996 ◽  
Vol 29 (4) ◽  
pp. 279-281 ◽  
Author(s):  
A. M. Borodin
Keyword(s):  
Normal Form ◽  
Finite Field ◽  
Jordan Normal Form ◽  
Triangular Matrices

A 3D Field Expression in Multilayered Structures using Jordan Normal Form

2012 ◽  
Vol 132 (8) ◽  
pp. 698-699 ◽  
Author(s):  
Hideaki Wakabayashi ◽  
Jiro Yamakita
Keyword(s):  
Normal Form ◽  
Jordan Normal Form ◽  
Multilayered Structures

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

Mr. Smith goes to Las Vegas: Randomized parallel computation of the Smith Normal form of polynomial matrices

1989 ◽  
pp. 317-322 ◽  
Author(s):  
Erich Kaltofen ◽  
M. S. Krishnamoorthy ◽  
B. David Saunders
Keyword(s):  
Normal Form ◽  
Parallel Computation ◽  
Las Vegas ◽  
Smith Normal Form ◽  
Polynomial Matrices

The Jordan normal form; decomposition theorem for modules

1964 ◽  
Vol 15 (1) ◽  
pp. 276-281 ◽  
Author(s):  
J. L. Brenner
Keyword(s):  
Normal Form ◽  
Decomposition Theorem ◽  
Jordan Normal Form

scholarly journals ROOT MULTIPLICITIES AND NUMBER OF NONZERO COEFFICIENTS OF A POLYNOMIAL

2007 ◽  
Vol 06 (03) ◽  
pp. 469-475 ◽  
Author(s):  
SANDRO MATTAREI
Keyword(s):  
Finite Field ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Similar Argument ◽  
Original Result ◽  
Exact Number ◽  
Nonzero Root ◽  
Root Multiplicities ◽  
Univariate Polynomial

It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate statement in positive characteristic. Furthermore, we present a new proof of the original result, which produces also the exact number of monic polynomials of a given degree for which the bound is attained. A similar argument allows us to determine the number of monic polynomials of a given degree, multiplicity of a given nonzero root, and number of nonzero coefficients, over a finite field of characteristic larger than the degree.

Algebraic properties of a digraph and its line digraph

2003 ◽  
Vol 04 (04) ◽  
pp. 377-393 ◽  
Author(s):  
C. Balbuena ◽  
D. Ferrero ◽  
X. Marcote ◽  
I. Pelayo
Keyword(s):  
Normal Form ◽  
Characteristic Polynomials ◽  
Jordan Normal Form ◽  
Line Digraph ◽  
Algebraic Properties ◽  
Adjacency Matrices ◽  
Number Of Cycles

Let G be a digraph, LG its line digraph and A(G) and A(LG) their adjacency matrices. We present relations between the Jordan Normal Form of these two matrices. In addition, we study the spectra of those matrices and obtain a relationship between their characteristic polynomials that allows us to relate properties of G and LG, specifically the number of cycles of a given length.

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