2,801 research outputs found
Polynomials with symmetric zeros
Polynomials whose zeros are symmetric either to the real line or to the unit
circle are very important in mathematics and physics. We can classify them into
three main classes: the self-conjugate polynomials, whose zeros are symmetric
to the real line; the self-inversive polynomials, whose zeros are symmetric to
the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric
by an inversion with respect to the unit circle followed by a reflection in the
real line. Real self-reciprocal polynomials are simultaneously self-conjugate
and self-inversive so that their zeros are symmetric to both the real line and
the unit circle. In this survey, we present a short review of these
polynomials, focusing on the distribution of their zeros.Comment: Keywords: Self-inversive polynomials, self-reciprocal polynomials,
Pisot and Salem polynomials, M\"obius transformations, knot theory, Bethe
equation
Salem-Boyd sequences and Hopf plumbing
Given a fibered link, consider the characteristic polynomial of the monodromy
restricted to first homology. This generalizes the notion of the Alexander
polynomial of a knot. We define a construction, called iterated plumbing, to
create a sequence of fibered links from a given one. The resulting sequence of
characteristic polynomials has the same form as those arising in work of Salem
and Boyd in their study of distributions of Salem and P-V numbers. From this we
deduce information about the asymptotic behavior of the large roots of the
generalized Alexander polynomials, and define a new poset structure for Salem
fibered links.Comment: 18 pages, 6 figures, to appear in Osaka J. Mat
String Reconstruction from Substring Compositions
Motivated by mass-spectrometry protein sequencing, we consider a
simply-stated problem of reconstructing a string from the multiset of its
substring compositions. We show that all strings of length 7, one less than a
prime, or one less than twice a prime, can be reconstructed uniquely up to
reversal. For all other lengths we show that reconstruction is not always
possible and provide sometimes-tight bounds on the largest number of strings
with given substring compositions. The lower bounds are derived by
combinatorial arguments and the upper bounds by algebraic considerations that
precisely characterize the set of strings with the same substring compositions
in terms of the factorization of bivariate polynomials. The problem can be
viewed as a combinatorial simplification of the turnpike problem, and its
solution may shed light on this long-standing problem as well. Using well known
results on transience of multi-dimensional random walks, we also provide a
reconstruction algorithm that reconstructs random strings over alphabets of
size in optimal near-quadratic time
Fixed Point Polynomials of Permutation Groups
In this paper we study, given a group of permutations of a finite set, the so-called fixed point polynomial , where is the number of permutations in which have exactly fixed points. In particular, we investigate how root location relates to properties of the permutation group. We show that for a large family of such groups most roots are close to the unit circle and roughly uniformly distributed round it. We prove that many families of such polynomials have few real roots. We show that many of these polynomials are irreducible when the group acts transitively. We close by indicating some future directions of this research. A corrigendum was appended to this paper on 10th October 2014. </jats:p
Structure of Certain Chebyshev-type Polynomials in Onsager's Algebra Representation
In this report, we present a systematic account of mathematical structures of
certain special polynomials arisen from the energy study of the superintegrable
-state chiral Potts model with a finite number of sizes. The polynomials of
low-lying sectors are represented in two different forms, one of which is
directly related to the energy description of superintegrable chiral Potts
\ZZ_N-spin chain via the representation theory of Onsager's algebra. Both two
types of polynomials satisfy some -term recurrence relations, and th
order differential equations; polynomials of one kind reveal certain
Chebyshev-like properties. Here we provide a rigorous mathematical argument for
cases , and further raise some mathematical conjectures on those
special polynomials for a general .Comment: 18 pages, Latex ; Typos corrected, Small changes for clearer
presentatio
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