Polynomial Analogue Research Articles

Polynomial Analogs of Theta Functions and Rational Solutions of the KP Equation

Abstract In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.

Generalised knotoids

Abstract In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

On the Two-Variable Analogue Matrix of Bessel Polynomials and Their Properties

In this paper, we explore a study focused on a two-variable extension of matrix Bessel polynomials. We initiate the discussion by introducing the matrix Bessel polynomials involving two variables and derive specific differential formulas and recurrence relations associated with them. Additionally, we present a segment detailing integral formulas for the extended matrix Bessel polynomials. Lastly, we introduce the Laplace–Carson transform for the two-variable matrix Bessel polynomial analogue.

A polynomial analogue of Berggren’s theorem on Pythagorean triples

Say that ( x , y , z ) (x, y, z) is a positive primitive integral Pythagorean triple if x , y , z x, y, z are positive integers without common factors satisfying x 2 + y 2 = z 2 x^2 + y^2 = z^2 . An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on ( 3 , 4 , 5 ) (3, 4, 5) and ( 4 , 3 , 5 ) (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren’s theorem in the context of a one-variable polynomial ring over a field of characteristic ≠ 2 \neq 2 . As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.

On the q-binomial identities involving the Legendre symbol modulo 3

We use a polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 to prove six identities involving the q-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

Counting pairs of polynomials whose product is a cube

Let Fq be the finite field with q elements. We consider the number of pairs (f1,f2) with f1f2 being a cube, where fi∈Fq[T] and deg⁡fi=n for i=1,2. By contour integration, we obtain an asymptotic formula in terms of either n or q. This is a polynomial analogue of a result for integers due to de la Bretèche, Kurlberg and Shparlinski [2].

On rational approximations of Poisson integrals on the interval by Fejer sums of Fourier – Chebyshev integral operators

Approximations of the Fejér sums of the Fourier – Chebyshev rational integral operators with restrictions on numerical geometrically different poles are herein studied. The object of research is the class of functions defined by Poisson integrals on the segment [–1, 1]. Integral representations of approximations and upper estimates of uniform approximations are established. In the case when the boundary function has a power singularity on the segment [–1, 1], upper estimates of pointwise and uniform approximations are found, and the asymptotic representation of the majorant of uniform approximations is found. As a separate problem, approximations of Poisson integrals for two geometrically different poles of the approximating rational function are considered. In this case, the optimal values of the parameters at which the highest rate of uniform approximations by the studied method is achieved are found. If the function |x|s, s ∈ (0, 1], is approximated, then this rate is higher than the corresponding polynomial analogues. Consequently, asymptotic expressions of the exact upper bounds of the deviations of Fejer sums of polynomial Fourier – Chebyshev series on classes of Poisson integrals on a segment are obtained. Estimates of uniform approximations by Fejer sums of polynomial Fourier – Chebyshev series of functions given by Poisson integrals on a segment with a boundary function having a power singularity are also obtained.

Exceptional Laurent biorthogonal polynomials through spectral transformations of generalized eigenvalue problems

AbstractA formulation is given for the spectral transformation of the generalized eigenvalue problem through the decomposition of the second‐order differential operators. This allows us to construct some Laurent biorthogonal polynomial systems with gaps in the degree of the polynomial sequence. These correspond to an exceptional‐type extension of the orthogonal polynomials, as an extension of the Laurent biorthogonal polynomials. Specifically, we construct the exceptional extension of the Hendriksen–van Rossum polynomials, which are biorthogonal analogs of the classical orthogonal polynomials. Similar to the cases of exceptional extensions of classical orthogonal polynomials, both state‐deletion and state‐addition occur.

Set-valued tableaux rule for Lascoux polynomials

Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Schützenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial description for Lascoux polynomials involving right keys and set-valued tableaux. Our description generalizes the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our description, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw.Mathematics Subject Classifications: 05E05Keywords: Lascoux polynomials, set-valued tableaux, crystal operators

Covering by Planks and Avoiding Zeros of Polynomials

Abstract We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball, we show that for every real nonzero $d$-variate polynomial $P$ of degree $n$, there exists a point in the unit $d$-dimensional ball at distance at least $1/n$ from the zero set of the polynomial $P$. Using the polynomial approach, we also prove the strengthening of the Fejes Tóth zone conjecture on covering a sphere by spherical segments, closed parts of the sphere between two parallel hyperplanes. In particular, we show that the sum of angular widths of spherical segments covering the whole sphere is at least $\pi $.

Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive

Motivated by the study of Macdonald polynomials, J. Haglund and A. Wilson introduced a nonsymmetric polynomial analogue of the chromatic quasisymmetric function called the chromatic nonsymmetric polynomial of a Dyck graph. We give a positive expansion for this polynomial in the basis of fundamental slide polynomials using recent work of Assaf-Bergeron on flagged ( P , ρ ) (P,\rho ) -partitions. We then derive the known expansion for the chromatic quasisymmetric function of Dyck graphs in terms of Gessel’s fundamental basis by taking a backstable limit of our expansion.

The limit empirical spectral distribution of complex matrix polynomials

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree k under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with k constant and (2) [Formula: see text] with n bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on k (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

Polynomial Analogue of Gandy’s Fixed Point Theorem

The paper suggests a general method for proving the fact whether a certain set is p-computable or not. The method is based on a polynomial analogue of the classical Gandy’s fixed point theorem. Classical Gandy’s theorem deals with the extension of a predicate through a special operator ΓΦ(x)Ω∗ and states that the smallest fixed point of this operator is a Σ-set. Our work uses a new type of operator which extends predicates so that the smallest fixed point remains a p-computable set. Moreover, if in the classical Gandy’s fixed point theorem, the special Σ-formula Φ(x¯) is used in the construction of the operator, then a new operator uses special generating families of formulas instead of a single formula. This work opens up broad prospects for the application of the polynomial analogue of Gandy’s theorem in the construction of new types of terms and formulas, in the construction of new data types and programs of polynomial computational complexity in Turing complete languages.

On [formula omitted]-analogs of descent and peak polynomials

Descent polynomials and peak polynomials, which enumerate permutations π∈Sn with given descent and peak sets respectively, have recently received considerable attention (Billey et al., 1989;Diaz-Lopez et al., 2019). We give several formulas for q-analogs of these polynomials which refine the enumeration by the length of π. In the case of q-descent polynomials we prove that the coefficients in one basis are stronglyq-log concave, and conjecture this property in another basis. For peaks, we prove that the q-peak polynomial is palindromic in q, resolving a conjecture of Diaz-Lopez et al. (2021).

Approximations on Classes of Poisson Integrals by Fourier–Chebyshev Rational Integral Operators

Introducing some classes of the functions defined by Poisson integrals on the segment $ [-1,1] $ and studying approximations by Fourier–Chebyshev rational integral operators on the classes, we establish integral expressions for approximations and upper bounds for uniform approximations. In the case of boundary functions with a power singularity on $ [-1,1] $ , we find the upper bounds for pointwise and uniform approximations and an asymptotic expression for a majorant of uniform approximations in terms of rational functions with a fixed number of prescribed geometrically distinct poles. Considering two geometrically distinct poles of the approximant of even multiplicity, we obtain asymptotic estimates for the best uniform approximation by this method with a higher convergence rate than polynomial analogs.

Combinatorial Interpretations of Lucas Analogues of Binomial Coefficients and Catalan Numbers

The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with {n}. It is then natural to ask if the resulting rational function is actually a polynomial in s and t with nonnegative integer coefficients and, if so, what it counts. The first simple combinatorial interpretation for this polynomial analogue of the binomial coefficients was given by Sagan and Savage, although their model resisted being used to prove identities for these Lucasnomials or extending their ideas to other combinatorial sequences. The purpose of this paper is to give a new, even more natural model for these Lucasnomials using lattice paths which can be used to prove various equalities as well as extending to Catalan numbers and their relatives, such as those for finite Coxeter groups.

Jeśmanowicz’ conjecture for polynomials

Let (a, b, c) be pairwise relatively prime integers such that $$a^2 + b^2 = c^2$$ . In 1956, Jeśmanowicz conjectured that the only solution of $$a^x + b^y = c^z$$ in positive integers is $$(x,y,z)=(2,2,2)$$ . In this note we prove a polynomial analogue of this conjecture.

Properties of Multivariate $${\varvec{b}}$$-Ary Stern Polynomials

Given an integer base $$b\ge 2$$, we investigate a multivariate b-ary polynomial analogue of Stern’s diatomic sequence which arose in the study of hyper b-ary representations of integers. We derive various properties of these polynomials, including a generating function and identities that lead to factorizations of the polynomials. We use some of these results to extend an identity of Courtright and Sellers on the b-ary Stern numbers $$s_b(n)$$. We also extend a result of Defant and a result of Coons and Spiegelhofer on the maximal values of $$s_b(n)$$ within certain intervals.

Q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].

Bethe Subalgebras in Braided Yangians and Gaudin-Type Models

In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.