22 research outputs found
Primitive Polynomials, Singer Cycles, and Word-Oriented Linear Feedback Shift Registers
Using the structure of Singer cycles in general linear groups, we prove that
a conjecture of Zeng, Han and He (2007) holds in the affirmative in a special
case, and outline a plausible approach to prove it in the general case. This
conjecture is about the number of primitive -LFSRs of a given order
over a finite field, and it generalizes a known formula for the number of
primitive LFSRs, which, in turn, is the number of primitive polynomials of a
given degree over a finite field. Moreover, this conjecture is intimately
related to an open question of Niederreiter (1995) on the enumeration of
splitting subspaces of a given dimension.Comment: Version 2 with some minor changes; to appear in Designs, Codes and
Cryptography
Enumeration of Linear Transformation Shift Registers
We consider the problem of counting the number of linear transformation shift
registers (TSRs) of a given order over a finite field. We derive explicit
formulae for the number of irreducible TSRs of order two. An interesting
connection between TSRs and self-reciprocal polynomials is outlined. We use
this connection and our results on TSRs to deduce a theorem of Carlitz on the
number of self-reciprocal irreducible monic polynomials of a given degree over
a finite field.Comment: 16 page
Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields
The probability for two monic polynomials of a positive degree n with
coefficients in the finite field F_q to be relatively prime turns out to be
identical with the probability for an n x n Hankel matrix over F_q to be
nonsingular. Motivated by this, we give an explicit map from pairs of coprime
polynomials to nonsingular Hankel matrices that explains this connection. A
basic tool used here is the classical notion of Bezoutian of two polynomials.
Moreover, we give simpler and direct proofs of the general formulae for the
number of m-tuples of relatively prime polynomials over F_q of given degrees
and for the number of n x n Hankel matrices over F_q of a given rankComment: 10 pages; to appear in the Journal of Combinatorial Theory, Series
Knots, perturbative series and quantum modularity
We introduce an invariant of a hyperbolic knot which is a map \alpha\mapsto
\mathbf{Phi}_\a(h) from to matrices with entries in
and with rows and columns indexed by the boundary
parabolic representations of the fundamental group of the
knot. These matrix invariants have a rich structure: (a) their
entry, where is the trivial and the
geometric representation, is the power series expansion of the Kashaev
invariant of the knot around the root of unity as an
element of the Habiro ring, and the remaining entries belong to generalized
Habiro rings of number fields; (b) the first column is given by the
perturbative power series of Dimofte--Garoufalidis; (c)~the columns of
are fundamental solutions of a linear -difference equation;
(d)~the matrix defines an -cocycle in
matrix-valued functions on that conjecturally extends to a smooth
function on and even to holomorphic functions on suitable complex
cut planes, lifting the factorially divergent series to
actual functions. The two invariants and are
related by a refined quantum modularity conjecture which we illustrate in
detail for the three simplest hyperbolic knots, the , and
pretzel knots. This paper has two sequels, one giving a different realization
of our invariant as a matrix of convergent -series with integer coefficients
and the other studying its Habiro-like arithmetic properties in more depth.Comment: 97 pages, 8 figure
Block companion Singer cycles, primitive recursive vector sequences, and coprime polynomial pairs over finite fields
We discuss a conjecture concerning the enumeration of nonsingular matrices over a finite field that are block companion and whose order is the maximum possible in the corresponding general linear group. A special case is proved using some recent results on the probability that a pair of polynomials with coefficients in a finite field is coprime. Connection with an older problem of Niederreiter about the number of splitting subspaces of a given dimension are outlined and an asymptotic version of the conjectural formula is established. Some applications to the enumeration of nonsingular Toeplitz matrices of a given size over a finite field are also discussed. (C) 2011 Elsevier Inc. All rights reserved