Sequence Of Laurent Polynomials Research Articles

Desingularization in the q-Weyl algebra

In this paper, we study the desingularization problem in the first q-Weyl algebra. We give an order bound for desingularized operators, and thus derive an algorithm for computing desingularized operators in the first q-Weyl algebra. Moreover, an algorithm is presented for computing a generating set of the first q-Weyl closure of a given q-difference operator. As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator.

Orthogonal Laurent polynomials. A new algebraic approach

In this paper, some important algebraic aspects in the theory of orthogonal Laurent polynomials, such as the three-term recurrence relation, the Christoffel–Darboux identity or the Liouville–Ostrogradski formula, are revisited from the Riemann–Hilbert window. These topics are considered for general ordered Laurent polynomial sequences, and not only for the usual “balanced” cases. In addition, a connection with Szegö polynomials (orthogonal polynomials in the unit circle) is explored.

Proof of a stronger version of the AJ Conjecture for torus knots

For a knot $K$ in $S^3$, the $sl_2$-colored Jones function $J_K(n)$ is a sequence of Laurent polynomials in the variable $t$, which is known to satisfy non-trivial linear recurrence relations. The operator corresponding to the minimal linear recurrence relation is called the recurrence polynomial of $K$. The AJ conjecture \cite{Ga04} states that when reducing $t=-1$, the recurrence polynomial is essentially equal to the $A$-polynomial of $K$. In this paper we consider a stronger version of the AJ conjecture, proposed by Sikora \cite{Si}, and confirm it for all torus knots.

Anchoring and steering gaussian quadrature for positive definite strong moment functionals

Algorithms for the construction of block-formed orthogonal Laurent polynomial sequences and corresponding Gaussian quadratures associated with positive definite strong moment functionals are introduced and compared. Cost estimates and numerical examples are included.

Asymptotics of the colored Jones function of a knot

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the $n$th colored Jones polynomial at $e^{\a/n}$, when $\a$ is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1/n$ when $\a$ is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the $n$th colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when $\a$ is near $2 \pi i$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.

Fermionic structure in the sine-Gordon model: Form factors and null-vectors

The form factor bootstrap in integrable quantum field theory allows one to capture local fields in terms of infinite sequences of Laurent polynomials called ‘towers’. For the sine-Gordon model, towers are systematically described by fermions introduced some time ago by Babelon, Bernard and Smirnov. Recently the authors developed a new method for evaluating one-point functions of descendant fields, using yet another fermions which act on the space of local fields. The goal of this paper is to establish that these two fermions are one and the same object. This opens up a way for answering the longstanding question about how to identify precisely towers and local fields.

Optimizing Gaussian Quadrature for Positive Definite Strong Moment Functionals

Classical Gaussian quadrature is anchored in the space of real polynomials at polynomial degree 0 and is monotonically directed by successively increasing polynomial degree. The results of recent research, investigating weighing anchor and charting various courses in search of the best currents in the space of real Laurent polynomials, are presented. Standard error bound minimization anchoring and steering algorithms which utilize the moments of positive definite strong moment functionals to guide the construction of ordered orthogonal Laurent polynomial sequences are introduced.

Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases { 1 , z − 1 , z , z − 2 , z 2 , … } and { 1 , z , z − 1 , z 2 , z − 2 , … } . We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.

DIFFERENCE AND DIFFERENTIAL EQUATIONS FOR THE COLORED JONES FUNCTION

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman–Gelca–LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.

Ordered OLPS and CF Nth numerators

Definitions, theorems and examples are established for a general model of Laurent polynomial spaces and ordered orthogonal Laurent polynomial sequences, ordered with respect to ordered bases and orthogonal with respect to inner products · = L ∘ ⊙ decomposed into transition functional ⊙ and strong moment functional, or, more generally, sample functional L couplings. Under this formulation that is shown to subsume those in the existing literature, new fundamental results are produced, including necessary and sufficient conditions for ordered OLPS to be sequences of nth numerators of continued fractions, in contrast to the classical result concerning nth denominators which is shown to hold only in special cases.

A non-commutative formula for the colored Jones function

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern–Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin–Morton–Rozansky conjecture, first proved by Bar–Natan and the first author about 10 years ago. Our results complement recent work of Huynh and Le, who also give a non-commutative formulae for the colored Jones function of a knot, starting from a non-commutative formula for the R matrix of the quantum group \(U_{q}(\mathfrak{sl}_{2})\); see Huynh and Le (in math.GT/0503296).

Orthogonal Laurent polynomials

A Favard type theorem for special sequences of Laurent polynomials is proved. This result is used to establish the relation between T-fractions and orthogonal Laurent polynomials (OLPs). Applications are given to T-fractions for (quotients of) hypergeometric functions of type 2 F 1 and confluent forms. In the special case of 2 F 1( a,1; c; z) a weight function is given on the unit circle in C for the corresponding OLPs.