Laurent Polynomials Research Articles

Rational symbolic cubature rules over the first quadrant in a Cartesian plane

In this paper we introduce a new symbolic Gaussian formula for theevaluation of an integral over the first quadrant in a Cartesianplane, in particular with respect to the weight function$w(x)=\exp(-x^T x-1/x^T x)$, where $x=(x_1,x_2)^T\in \mathbb{R}^2_+$. Itintegrates exactly a class of homogeneous Laurent polynomials withcoefficients in the commutative field of rational functions in twovariables. It is derived using the connection between orthogonalpolynomials, two-point Padé approximants, and Gaussian cubatures.We also discuss the connection to two-point Padé-typeapproximants in order to establish symbolic cubature formulas ofinterpolatory type. Numerical examples are presented toillustrate the different formulas developed in the paper.

Generalized power series with a limited number of factorizations

Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen–Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.

Counting rooted spanning forests for circulant foliation over a graph

In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests $f(n)$ for the infinite family of graphs $H_n=H_n(G_1,G_2,\ldots,G_m)$ obtained as a circulant foliation over a graph $H$ on $m$ vertices with fibers $G_1,G_2,\ldots,G_m.$ Each fiber $G_i=C_n(s_{i,1},s_{i,2},\ldots,s_{i,k_i})$ of this foliation is the circulant graph on $n$ vertices with jumps $s_{i,1},s_{i,2},\ldots,s_{i,k_i}.$ This family includes the family of generalized Petersen graphs, $I$-graphs, sandwiches of circulant graphs, discrete torus graphs and others. The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form $f(n)=p f(H)a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the number of odd elements in the set of $s_{i,j}.$ Finally, we find an asymptotic formula for $f(n)$ through the Mahler measure of the associated Laurent polynomial.

The Derivation of Elastic Fields of a Curvilinear Inclusion

The disturbed elastic fields of a curvilinear inclusion in an isotropic elastic plane are investigated analytically by a newly proposed technique. The boundary of the inclusion is characterized by arbitrary Laurent polynomials in the 2D Cartesian coordinate system, and constant eigenstrains are considered to occur in the inclusion. Based on the irreducible decomposition of an arbitrary tensor, the Eshelby tensor is attributed to two integrals on the curved boundary of the inclusion. The analytical solutions for the induced stress and displacement fields outside the inclusion domain are explicitly derived by utilizing the newly developed technique, including the salient features of the Faber polynomials. Examples show the efficiency of the technique in this paper.

Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

In 1974, the author proved that the codimension of the ideal (g1,g2,…,gd) generated in the group algebra K[Zd] over a field K of characteristic 0 by generic Laurent polynomials having the same Newton polytope Γ is equal to d!×Volume(Γ). Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set Bsh of monomials of cardinality d!×Volume(Γ), whose equivalence classes form a basis of the quotient algebra K[Zd]/(g1,g2,…,gd). The set Bsh is constructed inductively from any shelling sh of the polytope Γ. Using the Bsh structure, we prove that the associated graded K -algebra grΓ(K[Zd]) constructed from the Arnold–Newton filtration of K -algebra K[Zd] has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials (f1,f2,…,fd) with the same Newton polytope Γ, the set Bsh defines a monomial basis of the quotient algebra K[Zd]/(g1,g2,…,gd).

Self-intersections of Laurent polynomials and the density of Jordan curves

We extend Quine’s bound on the number of self-intersection of curves with polynomial parameterization to the case of Laurent polynomials. As an application, we show that circle embeddings are dense among all maps from a circle to a plane with respect to an integral norm.

Hereditary atomicity in integral domains

If every subring of an integral domain is atomic, we say that the latter is hereditarily atomic. In this paper, we study hereditarily atomic domains. First, we characterize when certain direct limits of Dedekind domains are Dedekind domains in terms of atomic overrings. Then we use this characterization to determine the fields that are hereditarily atomic. On the other hand, we investigate hereditary atomicity in the context of rings of polynomials and rings of Laurent polynomials, characterizing the fields and rings whose rings of polynomials and rings of Laurent polynomials, respectively, are hereditarily atomic. As a result, we obtain two classes of hereditarily atomic domains that cannot be embedded into any hereditarily atomic field. By contrast, we show that rings of power series are never hereditarily atomic. Finally, we make some progress on the still open question of whether every subring of a hereditarily atomic domain satisfies ACCP.

Skein and cluster algebras of unpunctured surfaces for $${\mathfrak {sl}}_3$$

For an unpunctured marked surface \(\Sigma \), we consider a skein algebra \({\mathscr {S}}_{{\mathfrak {sl}}_{3},\Sigma }^{q}\) consisting of \({\mathfrak {sl}}_3\)-webs on \(\Sigma \) with the boundary skein relations at marked points. We construct a quantum cluster algebra \({\mathscr {A}}^q_{{\mathfrak {sl}}_3,\Sigma }\) inside the skew-field \(\text {Frac} {\mathscr {S}}_{{\mathfrak {sl}}_{3},\Sigma }^{q}\) of fractions, which quantizes the cluster \(K_2\)-structure on the moduli space \({\mathcal {A}}_{SL_3,\Sigma }\) of decorated \(SL_3\)-local systems on \(\Sigma \). We show that the cluster algebra \({\mathscr {A}}^q_{{\mathfrak {sl}}_3,\Sigma }\) contains the boundary-localized skein algebra \({\mathscr {S}}_{{\mathfrak {sl}}_{3},\Sigma }^{q}[\partial ^{-1}]\) as a subalgebra, and their natural structures, such as gradings and certain group actions, agree with each other. We also give an algorithm to compute the Laurent expressions of a given \({\mathfrak {sl}}_3\)-web in certain clusters and discuss the positivity of coefficients. In particular, we show that the bracelets and the bangles along an oriented simple loop in \(\Sigma \) have Laurent expressions with positive coefficients, hence give rise to quantum GS-universally positive Laurent polynomials.

A new class of finitely generated polynomial subalgebras without finite SAGBI bases

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras.

Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics

In the present paper, we investigate a family of circulant graphs with non-fixed jumpsGn= Cβn(s1, ... , sk, α1n, ... , αln), 1 ≤ s1< ... < sk≤ [βn/2], 1 ≤ α1< ... < αl ≤ [β/2]. Here n is an arbitrary large natural number and integers s1, ... , sk, α1, ... , αl are supposed to be fixed.First, we present an explicit formula for the number of spanning trees in the graph Gn. This formula is a product of βsk-1 factors, each given by the n-th Chebyshev polynomial of the first kind evaluated at the roots of some prescribed polynomial of degree sk. Next, we provide some arithmetic properties of the complexity function. We show that the number of spanning trees in Gn can be represented in the form τ(n) = p n a(n)2, where a(n) is an integer sequence and p is a prescribed natural number depending of parity of β and n. Finally, we find an asymptotic formula for τ(n) through the Mahler measure of the associated Laurent polynomials differing by a constant from 2k - ∑i = 1k(zsi+z−si).

Generic Newton polygon of the L-function of n variables of the Laurent polynomial I

Generic Newton polygon of the L-function of n variables of the Laurent polynomial I

THE STABLE LIMIT DAHA AND THE DOUBLE DYCK PATH ALGEBRA

AbstractWe study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/– stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation of the +stable limit DAHA.

Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

We study the growth and Gelfand-Kirillov dimension (GK-dimension) of the generalized Weyl algebra (GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and can be applied to determine the growth, GK-dimension, simplicity and cancellation properties of some GWAs.

Lower and upper bounds of condition number for Vandermonde‐wise matrices and method of fundamental solutions using pseudo radial‐lines

AbstractConsider the method of fundamental solutions (MFS) for 2D Laplace's equation in a bounded simply connected domain . In the standard MFS, the source nodes are located on a closed contour outside the domain boundary , which is called pseudo‐boundary. For circular, elliptic, and general closed pseudo‐boundaries, analysis and computation have been studied extensively. New locations of source nodes are proposed along two pseudo radial‐lines outside . Numerical results are very encouraging and promising. Since the success of the MFS mainly depends on stability, our efforts are focused on deriving the lower and upper bounds of condition number (Cond). The study finds stability properties of new Vandermonde‐wise matrices on nodes with . The Vandermonde‐wise matrix is called in this article if it can be decomposed into the standard Vandermonde matrix. New lower and upper bounds of Cond are first derived for the standard Vandermonde matrix, and then for new algorithms of the MFS using two pseudo radial‐lines. Both lower and upper bounds of Cond are intriguing in the stability study for the MFS. Numerical experiments are carried out to verify the stability analysis made. Since the fundamental solutions (as ) are the basis functions of the MFS, new Vandermonde‐wise matrices are found. Since the nodes with may come from approximations and interpolations by the Laurent polynomials with singular part, the conclusions in this article are important not only to the MFS but also to matrix analysis.

Polynomial and signature invariants for pseudo-links via Goeritz matrices

In this paper, we introduce the Goeritz matrix for a pseudo-link whose entries lie in the Laurent polynomial ring [Formula: see text], which generalizes the Goeritz matrix for a classical link. We show that the determinant of a modified Goeritz matrix gives a Laurent polynomial invariant for pseudo-links in one variable u with integer coefficients. We also introduce the notions of the signature, determinant, and nullity of pseudo-links. Further, we discuss some properties of the invariants and compute the polynomials and those numerical invariants for several pseudo-knot families.

The number of rooted forests in circulant graphs

In this paper, we develop a new method to produce explicit formulas for the number fG(n) of rooted spanning forests in the circulant graphs G = Cn(s1,s2,…,sk) and G = C2n(s1,s2,…,sk,n). These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form fG(n) = p a(n)2, where a(n) is an integer sequence and p is a certain natural number depending on the parity of n. Finally, we find an asymptotic formula for fG(n) through the Mahler measure of the associated Laurent polynomial P(z) = 2k + 1−∑i = 1k(zsi+z−si).

Characterization of Polynomials Whose Large Powers have Fully Positive Coefficients

We give a criterion which characterizes a real multi-variate Laurent polynomial with full-dimensional smooth Newton polytope to have the property that all sufficiently large powers of the polynomial have fully positive coefficients. Here a Laurent polynomial is said to have fully positive coefficients if the coefficients of its monomial terms indexed by the lattice points of its Newton polytope are all positive. Our result generalizes an earlier result of Colin Tan and the author, which corresponds to the special case when the Newton polytope of the Laurent polynomial is a translate of a standard simplex. The result also generalizes a result of De Angelis, which corresponds to the special case of univariate polynomials. As an application, we also give a characterization of certain polynomial spectral radius functions of the defining matrix functions of Markov chains.

Generalized Lucas congruences and linear p-schemes

We observe that a sequence satisfies Lucas congruences modulo p if and only if its values modulo p can be described by a linear p-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests natural generalizations of the notion of Lucas congruences. To illustrate this point, we prove explicit generalized Lucas congruences for integer sequences that can be represented as the constant terms of P(x,y)nQ(x,y) where P and Q are certain Laurent polynomials.

Irreducible integrable modules for the full toroidal Lie algebras coordinated by rational quantum torus

Let Cq be the non-commutative Laurent polynomial ring associated with a (n+1)×(n+1) rational quantum matrix q. Let sld(Cq)⊕HC1(Cq) be the universal central extension of Lie subalgebra sld(Cq) of gld(Cq). We take the Lie algebra τ=gld(Cq)⊕HC1(Cq). Let Der(Cq) be the Lie algebra of all derivations of Cq and consider the Lie algebra τ˜=τ⋊Der(Cq), called the full toroidal Lie algebra coordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for τ˜ with nonzero central action on the modules.

A homological model for Uq𝔰𝔩2 Verma modules and their braid representations

We extend Lawrence's representations of the braid groups to relative homology modules, and we show that they are free modules over a Laurent polynomials ring. We define homological operators and we show that they actually provide a representation for an integral version for $U_q \mathfrak{sl}(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules, and we show it to preserve the integral ring of coefficients, the action of $U_q \mathfrak{sl}(2)$, the braid group representations and their grading. This recovers an integral version for Kohno's theorem relating absolute Lawrence representations with quantum braid representation on highest weight vectors. It is an extension of the latter theorem as we get rid of generic conditions on parameters, and as we recover the entire product of Verma-modules as a braid group and a $U_q \mathfrak{sl}(2)$-module.