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Erdös–Lax inequality relates the sup norm of the derivative of a polynomial along the unit circle to that of the polynomial itself (on the unit circle). This paper aims to extend the classical Erdös–Lax inequality to the polar derivative of a polynomial by using the extreme coefficients of the given polynomial. The obtained results not only enrich the realm of Erdös–Lax-type inequalities but also offer a promising avenue for diverse applications where these inequalities play a pivotal role. To illustrate the practical significance of our results, we present a numerical example. It vividly demonstrates that our bounds are considerably sharper than the existing ones in the extensive literature on this captivating subject. [ABSTRACT FROM AUTHOR]

We establish some -inequalities for the polar derivative of a polynomial. Our results generalize some known Bernstein-type polynomial inequalities and include as special cases several interesting generalizations of some Zygmund-type inequalities for polynomials as well. [ABSTRACT FROM AUTHOR]

We prove that if P( z) is a polynomial of degree n with zeros z , that satisfy $${|z_m| \geq K_m \geq 1}$$ , $${1 \leq m \leq n}$$ , then for any p > 0, and for every complex number α, with $${|\alpha| \geq 1}$$ , we have where $${G_{p}=\big\{\frac{2\pi}{\int_0^{2\pi}|t_0+e^{i\theta}|^{p}\,d\theta}\big\}^{{1}/{p}}}$$ , and $${t_0=\big\{1+\frac{n}{\sum_{m=1}^n\frac{1}{K_m-1}}\big\}}$$ if $${K_{m}>1}$$ $${(1\leq m \leq n)}$$ , and t = 1 if K = 1 for some m, $${1\leq m\leq n}$$ . Our results generalize and sharpen several of the known results. [ABSTRACT FROM AUTHOR]

All the roots of the general nth degree trinomial admit certain convenient representations in terms of the Lambert and Euler series for the asymmetric and symmetric cases of the trinomial equation, respectively. Previously, various methods have been used to provide the proofs for the general terms of these two series. Taking n to be any real or complex number, we presently give an alternative proof using the Bell (or exponential) polynomials. The ensuing series is summed up yielding a single, compact, explicit, analytical formula for all the trinomial roots as the confluent Fox-Wright function 1Ψ1. Moreover, we also derive a slightly different, single formula of the trinomial root raised to any power (real or complex number) as another 1Ψ1 function. Further, in this study, the logarithm of the trinomial root is likewise expressed through a single, concise series with the binomial expansion coefficients or the Pochhammer symbols. These findings are anticipated to be of considerable help in various applications of trinomial roots. Namely, several properties of the 1Ψ1 function can advantageously be employed for its implementations in practice. For example, the simple expressions for the asymptotic limits of the 1Ψ1 function at both small and large values of the independent variable can be used to readily predict, by analytical means, the critical behaviors of the studied system in the two extreme conditions. Such limiting situations can be e.g. at the beginning of the time evolution of a system, and in the distant future, if the independent variable is time, or at low and high doses when the independent variable is radiation dose, etc. The present analytical solutions for the trinomial roots are numerically illustrated in the genome multiplicity corrections for survival of synchronous cell populations after irradiation. [ABSTRACT FROM AUTHOR]

Let p(z) be a polynomial of degree n. The polar derivative of p(z) with respect to a complex number α is defined by D α p (z) = n p (z) + (α - z) p ′ (z). If p (z) = z s ∑ j = 0 n - s c j z j , 0 ≤ s ≤ n , has all its zeros in | z | ≤ k , k ≥ 1 , then for | α | ≥ k , Kumar and Dhankhar [Bull, Math. Soc. Sci. Math., 63(4), 359-367 (2020)] proved max | z | = 1 | D α p (z) | ≥ n (| α | - k) 1 + k n - s 1 + (| c n - s | k n - | c 0 | k s) (k - 1) 2 (| c n - s | k n + | c 0 | k s + 1) max | z | = 1 | p (z) |. In this paper, we first improve the above inequality. Besides, we are able to prove an improvement of a result due to Govil and Mctume [Acta. Math. Hungar., 104, 115-126 (2004)] and also prove an inequality for a subclass of polynomials having no zero in | z | < k , k ≤ 1 . [ABSTRACT FROM AUTHOR]

Let $$P(z)=a_{0}+\sum _{\nu =\mu }^{n}a_{\nu }z^{\nu },~1\le \mu \le n,$$ be a polynomial of degree $$n$$ such that $$P(z)\ne 0$$ in $$|z|< k,~k> 0,$$ then for $$01$$ by proving and In this paper, we obtain improvements of the above inequalities by involving some of the coefficients of the polynomial $$P(z).$$ Besides this, our results also generalize and refines the results of Barchand and Dewan (J Math Anal Appl 336:171-179, ). [ABSTRACT FROM AUTHOR]

The article discusses the fundamental theorem of algebra. It mentions that every non-constant single-variable polynomial with complex coefficients has at least one complex root. It includes polynomials with real coefficients, since every real number is a complex number with an imaginary part equal to zero. It adds that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots.

Let p (z) be a polynomial of degree n having no zero in | z | < k , k ≤ 1 , then Govil [Proc. Nat. Acad. Sci., 50(1980), 50-52] proved max | z | = 1 | p ′ (z) | ≤ n 1 + k n max | z | = 1 | p (z) | , provided | p ′ (z) | and | q ′ (z) | attain their maxima at the same point on the circle | z | = 1 , where q (z) = z n p (1 z ‾) ‾. In this paper, we present integral mean inequalities of Turán- and Erdös-Lax-type for the polar derivative of a polynomial by involving some coefficients of the polynomial, which refine some previously proved results and one of our results improves the above Govil inequality as a special case. These results incorporate the placement of the zeros and some coefficients of the underlying polynomial. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to some previously established results. [ABSTRACT FROM AUTHOR]

Let $\mathscr{F}$ be a family of solutions of Laplace equations in a domain D and for each $f\in\mathscr{F}$, f has only zeros of multiplicity at least k. Let n be a positive integer and such that $n\geq\frac{1+\sqrt{1+4k(k+1)^{2}}}{2k}$. Let a be a complex number such that $a\neq0$. If for each pair of functions f and g in $\mathscr{F}$, $f^{n}f^{(k)}$ and $g^{n}g^{(k)}$ share a value in D, then $\mathscr{F}$ is normal in D. [ABSTRACT FROM AUTHOR]

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing t = 0 Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of U q ′ (sl ^ 2) -modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials. [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain a q-analogue for partial-fraction decomposition of the rational function x M (x + 1) n λ . As applications, we deduce the corresponding q-algebraic and q-combinatorial identities which are the q-extensions of Chu' results. [ABSTRACT FROM AUTHOR]

This present work provides the initial co-efficient bounds of the function f(z) which is defined in open unit disk D . We introduced bi-univalent class C Σ (λ , t , ν) . Making use of Horadam polynomials h n (ν) and the generating function Π (ν , z) , we estimated the bounds of a 2 and a 3 for the given function to be in the defined class. Moreover Fekete-Szego inequalities are calculated. In addition to all the results obtained mathematically, we provide application for Horadam polynomial in computer vision. [ABSTRACT FROM AUTHOR]

We suggest an ansatz for representation of affine Yangian Y gl ̂ 1 by differential operators in the triangular set of time-variables Pa,i with 1 ⩽ i ⩽ a, which saturates the MacMahon formula for the number of 3d Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting "cut-and-join" generators ψn of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods. [ABSTRACT FROM AUTHOR]

The paper continues the research of R. Kronover and J. Minlor, H.-O. Peitgen and P. H. Richter. Using mathematical methods and computer experiments, the first- and second-order framings of the Mandelbrot sets of three families of the quadratic polynomials of a complex variable are revealed. The connection between the first- and second-order framings of Mandelbrot sets of the functions f2(z) = z2 + cz, f3(z) = z2 + z + c, and f4(z) = cz2 + c with remarkable curves (cardioid, lemniscate, and circle) was established. The algorithms of constructing the framings of Mandelbrot sets of considered functions in the mathematical package MathCad and Pascal programs have been worked out. Algorithms for constructing Mandelbrot sets in Pascal programs are developed. [ABSTRACT FROM AUTHOR]

We study the problem of describing the triples , , where is the (co)metric associated with a symmetric second-order differential operator defined on a domain of and such that there exists an orthonormal basis of consisting of polynomials which are eigenvectors of and this basis is compatible with the filtration of the space of polynomials by some weighted degree. In a joint paper of D. Bakry, M. Zani, and the author this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2 but for a weighted degree with arbitrary positive weights. [ABSTRACT FROM AUTHOR]

Shapovalov elements of quantum groups are special polynomials in negative simple root vectors with coefficients in the rational Cartan subalgebra that relate singular vectors in reducible Verma modules with their highest vectors. We give explicit expressions for Shapovalov elements of nonexceptional quantum groups in terms of matrix elements of quantum -operators using calculations on Hasse diagrams associated with auxiliary representations. [ABSTRACT FROM AUTHOR]

Stenger conjectures are claims about the location of the eigenvalues of matrices whose elements are certain integrals involving basic Lagrange interpolating polynomials supported on the zeros of orthogonal polynomials. In this paper, we show the validity of the extended Stenger conjecture for families of classical orthogonal polynomials. We also show the validity of the restricted Strenger conjecture for a family of Jacobi and generalized Laguerre orthogonal polynomials. A connection with the A-stability of the collocation Runge-Kutta methods is investigated. [ABSTRACT FROM AUTHOR]

We obtain the characteristic polynomials and a number of spectral-based indices such as the Riemann-Zeta functional indices and spectral entropies of n-dimensional hypercubes using recursive Hadamard transforms. The computed numerical results are constructed for up to 23-dimensional hypercubes. While the graph energies exhibit a J-curve as a function of the dimension of the n-cubes, the spectra-based entropies exhibit a linear dependence on the dimension. We have also provided structural interpretations for the coefficients of the characteristic polynomials of n-cubes and obtain expressions for the integer sequences formed by the spectral-based Riemann-Zeta functions. [ABSTRACT FROM AUTHOR]

Certain triples of power series, considered by I. Macdonald, give a natural framework for many combinatorial and number theoretic sequences, such as the Stirling, Bernoulli and harmonic numbers and partitions of different kinds. The power series in such a triple are closely linked by identities coming from the theory of symmetric functions. We extend the work of Z.-H. Sun, who developed similar ideas, and Macdonald, revealing more of the structure of these triples. De Moivre polynomials play a key role in this study. [ABSTRACT FROM AUTHOR]

We revisit the classical subject of equidistribution of the roots of Littlewood-type polynomials. More precisely, we show that the roots of the family of polynomials Ψk(z) = zk−zk−1−⋯−1, k ⩾ 1, are uniformly distributed around the unit circle in the strong quantitative form, confirming a conjecture from [C.-A. Gómez and F. Luca, Commentat. Math. Univ. Carol., On the distribution of roots of zk − zk−1−⋯−z − 1, 62(3):291–296, 2021]. [ABSTRACT FROM AUTHOR]

We discover a "no semistability at infinity" phenomenon for complete Calabi-Yau metrics asymptotic to cones, which is proved by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson and the first author. It is in sharp contrast to the setting of local singularities of Kähler-Einstein metrics. A byproduct of the proof is a polynomial convergence rate to the asymptotic cone for such manifolds, which bridges the gap between the general theory of Colding-Minicozzi and the classification results of Conlon-Hein. [ABSTRACT FROM AUTHOR]

In this paper, we combine the operators N and D α , and investigate the dependence of N [ D α (P (R z) - β P (r z) ] on the maximum modulus of P(z) on | z | = 1 for every real or complex numbers α and β with | α | ≥ 1 , | β | ≤ 1 and R > r ≥ 1 . The present results include some known polynomial inequalities as special cases. [ABSTRACT FROM AUTHOR]

In recent years, many authors studied the arithmetic, analytic and geometric aspects of roots of characteristic polynomials of some linear recurrence sequences. Many of these polynomials are particular cases of the three-parameter family f k , p , q (x) = x k - p x k - 1 - q x - 1 , for non-negative integers k, p and q, with k ≥ 3 and p ≥ 1 . In this work, we prove some properties related to the roots of the polynomial f k , p , q (x) which, for instance, make their applications possible in a large class of Diophantine problems. [ABSTRACT FROM AUTHOR]

We present new closed-form formulas for the matrix logarithm. Our method is direct and elementary, and it gives tractable and manageable formulas not current in the extensive literature on this essential subject. Using these results, an elegant explicit formula for the principal matrix logarithm can be obtained. Several special cases and examples are stated to illustrate the method presented in this paper. [ABSTRACT FROM AUTHOR]

Recutting is an operation on planar polygons defined by cutting a polygon along a diagonal to remove a triangle, and then reattaching the triangle along the same diagonal but with opposite orientation. Recuttings along different diagonals generate an action of the affine symmetric group on the space of polygons. We show that this action is given by cluster transformations and is completely integrable. The integrability proof is based on interpretation of recutting as refactorization of quaternionic polynomials. [ABSTRACT FROM AUTHOR]

We study the problem of approximating the cone of positive semidefinite (PSD) matrices with a cone that can be described by smaller-sized PSD constraints. Specifically, we ask the question: "how closely can we approximate the set of unit-trace n × n PSD matrices, denoted by D, using at most N number of k × k PSD constraints?" In this paper, we prove lower bounds on N to achieve a good approximation of D by considering two constructions of an approximating set. First, we consider the unit-trace n × n symmetric matrices that are PSD when restricted to a fixed set of k-dimensional subspaces in R n . We prove that if this set is a good approximation of D, then the number of subspaces must be at least exponentially large in n for any k = o (n) . Second, we show that any set S that approximates D within a constant approximation ratio must have superpolynomial S + k -extension complexity. To be more precise, if S is a constant factor approximation of D, then S must have S + k -extension complexity at least exp (C · min { n , n / k }) where C is some absolute constant. In addition, we show that any set S such that D ⊆ S and the Gaussian width of S is at most a constant times larger than the Gaussian width of D must have S + k -extension complexity at least exp (C · min { n 1 / 3 , n / k }) . These results imply that the cone of n × n PSD matrices cannot be approximated by a polynomial number of k × k PSD constraints for any k = o (n / log 2 n) . These results generalize the recent work of Fawzi (Math Oper Res 46(4):1479–1489, 2021) on the hardness of polyhedral approximations of S + n , which corresponds to the special case with k = 1 . [ABSTRACT FROM AUTHOR]

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function x M (x + 1) n λ (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu' results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

Let N p denote the Newton map induced by a complex polynomial p. Schleicher (Ergod Theory Dyn Syst 22:935–945, 2002) showed that there exists a uniform bound A d , such that for every polynomial p of degree d and for every point z 0 in the immediate basin of a root α of p, we have | z 0 - α | ≤ A d | N p (z 0) - z 0 | . Schleicher also proved that A d ≤ f d , where f d = d 2 (d - 1) 2 (2 d - 1) 2 d d ∼ 4 d - 1 d 2 π d . In 2020, the authors showed that A d < 3 d (3.02) d if d ≥ 12. The goal of this paper is to establish a better bound for A d using collaboration between roots of p. We establish that A d < (1.77) d for all sufficiently large d. As a consequence, it gives a better bound on the expected total number of iterations of Newton's method required to reach all roots of every polynomial p within a given precision. [ABSTRACT FROM AUTHOR]

The Erdős–Lax Theorem states that, if P(z) is a polynomial of degree n having no zeros in | z | < 1 , then max | z | = 1 | P ′ (z) | ≤ n 2 max | z | = 1 | P (z) |. The problem of generalizing the Erdős–Lax Theorem to the class of polynomials having no zeros in | z | < K , K ≤ 1 is still not settled. Motivated by the above open problem, in this paper we prove some inequalities for a class of polynomials having no zeros in | z | < K , K ≤ 1 , and all these zeros lie either on a ray L emanating from the origin or zeros are symmetrically placed along this line L. Several important consequences are also discussed. Our results are the best possible and examples for the equality cases have been presented. [ABSTRACT FROM AUTHOR]

Extending the classical result that the roots of a polynomial with coefficients in C are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if F = { f λ : λ ∈ Λ } is a set of polynomials in C [ t 1 , ... , t n ] and if is a set of polynomials in such that g λ is an infinitesimal deformation of f λ for all λ ∈ Λ , then the nonstandard affine variety is an infinitesimal deformation of the affine variety V (F). [ABSTRACT FROM AUTHOR]

Let Φ n (k) (x) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values Φ n (k) (1) for fixed positive integers n. D. H. Lehmer proved that Φ n (k) (1) / Φ n (1) is a polynomial of the Euler totient function ϕ (n) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that Φ n (k) (1) / Φ n (1) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: 2 Φ n (3) (1) is divisible by ϕ (n) - 2 . Moreover, if k is greater than 1, then Φ n (2 k + 1) (1) is divisible by ϕ (n) - 2 k . The proof depends on a new combinatorial identity for general self-reciprocal polynomials over Z , which gives rise to a formula that expresses the value Φ n (k) (1) as a Z -linear combination of the coefficients in the minimal polynomial of 2 cos (2 π / n) - 2 . As a supplement, we show the monotonic increasing property of Φ n (x) on [ 1 , ∞) in two ways. [ABSTRACT FROM AUTHOR]

We study the 'critical moments' of subcritical Gaussian multiplicative chaos (GMCs) in dimensions d ≤ 2 . In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale. This is verified for an integer case in the setting of circular unitary ensemble, extending and strengthening the results of Claeys et al. and Fahs to higher-order moments. [ABSTRACT FROM AUTHOR]

We establish some inequalities for the modulus of the derivative of rational functions of Bernstein and Turán type in the sup-norm on the unit disk in the complex plane. These results produce some sharper inequalities while taking into account the placement of zeros of the underlying rational function. Moreover, many inequalities for polynomials and polar derivatives follow as special cases as well. [ABSTRACT FROM AUTHOR]

In this paper we give a classification of the asymptotic expansion of the q-expansion of reciprocals of Eisenstein series E k of weight k for the modular group SL 2 (Z) . For k ≥ 12 even, this extends results of Hardy and Ramanujan, and Berndt, Bialek, and Yee, utilizing the Circle Method on the one hand, and results of Petersson, and Bringmann and Kane, developing a theory of meromorphic Poincaré series on the other. We follow a uniform approach, based on the zeros of the Eisenstein series with the largest imaginary part. These special zeros provide information on the singularities of the Fourier expansion of 1 / E k (z) with respect to q = e 2 π i z . [ABSTRACT FROM AUTHOR]

It turns out that the partial sums g n z = ∑ k = 0 n a 1 k ⋯ a p k b 1 k ⋯ b q k z k k ! of a generalized hypergeometric series pFq(a1,... , ap; b1,... , bq; z) with parameters aj, bl ∈ ℂ\{0,−1,−2,...} are orthogonal Sobolev polynomials. The corresponding monic polynomials Gn(z) are RI-type polynomials. Therefore, they are related to biorthogonal rational functions. The polynomials gn satisfy a differential equation (in z) and a recurrence relation (in n). We study the integral representations for gn and some of their basic properties. It is shown that partial sums of any power series with nonzero coefficients are also related to biorthogonal rational functions. We also establish a relationship between the polynomials gn(z) and the Jacobi-type pencils and their associated polynomials. [ABSTRACT FROM AUTHOR]

Given a complex polynomial p(z) of degree n, we are interested in characterization and computation of local optima — maxima or minima — of |p| over a disc D of radius r centered at the origin, or its boundary, ∂D. From the maximum modulus principle (MMP), together with parametrization of ∂D, maximization reduces to a univariate optimization. However, reduction to univariate optimization does not guarantee efficiency in the computation of local optima, nor gives any specific characterization, nor provides a bound on the number of local optima. In this article, we utilize the geometric modulus principle, a stronger result than MMP, to prove novel characterizations of local optima over D or ∂D. On the one hand, we prove any local optimum that lies on ∂D is necessarily a fixed point of F r (z) = ± r (p / p ′) / | p / p ′ | and also give a sufficiency condition. On the other hand, we show any fixed point is necessarily a root of a derived polynomialPr(z) of degree 2n defined in terms of p, p ′ and their conjugate reciprocals. This also proves the number of local optima is at most 2n, an upper bound that can be attained. Our results imply that all desired local optima can be approximated to arbitrary precision in polynomial time. In particular, the classic ∥ p ∥ ∞ can be computed efficiently. Implementation of our algorithm using Matlab and MPSolve is capable of computing all roots of Pr(z) of degree up to 250 and subsequently all local optima of |p| to precision of 10− 13 in less than 3 s. In summary, the proposed theoretical algorithm is also practical and efficient. Additionally, based on the derived formulas we propose specialized algorithms and present sample visualization through polynomiography. [ABSTRACT FROM AUTHOR]

Certain orthogonal polynomials are involved in various problems occurring in probability, celestial mechanics, combinatorics, and other fields of theoretical and applied sciences. The article aims to introduce -Sheffer sequences of Bessel type using -Riordan matrices by means of the Eulerian generating function. Series expansion and determinant expressions are obtained for these -sequences. We discuss the orthogonality of -Sheffer sequences of Bessel type by establishing their three term -recurrence relations. As applications, we obtain the Eulerian generating functions and other properties for some special -Sheffer sequences of Bessel type including the continuous -Hermite and -exponential polynomials of Bessel type. [ABSTRACT FROM AUTHOR]

Let P ∈ Z [ X , Y ] be a given square-free polynomial of total degree d with integer coefficients of bitsize less than τ , and let V R (P) : = { (x , y) ∈ R 2 ∣ P (x , y) = 0 } be the real planar algebraic curve implicitly defined as the vanishing set of P. We give a deterministic algorithm to compute the topology of V R (P) in terms of a simple straight-line planar graph G that is isotopic to V R (P) . The upper bound on the bit complexity of our algorithm is in O ~ (d 5 τ + d 6) (The expression "the complexity is in O ~ (f (d , τ)) " with f a polynomial in d , τ is an abbreviation for the expression "there exists a positive integer c such that the complexity is in O ((log d log τ) c f (d , τ)) "); which matches the current record bound for the problem of computing the topology of a planar algebraic curve. However, compared to existing algorithms with comparable complexity, our method does not consider any change of coordinates, and more importantly the returned simple planar graph G yields the cylindrical algebraic decomposition information of the curve in the original coordinates. Our result is based on two main ingredients: First, we derive amortized quantitative bounds on the roots of polynomials with algebraic coefficients as well as adaptive methods for computing the roots of bivariate polynomial systems that actually exploit this amortization. The results we obtain are more general that the previous literature. Our second ingredient is a novel approach for the computation of the local topology of the curve in a neighborhood of all singular points. [ABSTRACT FROM AUTHOR]

In this paper, we study the sparsity-adapted complex moment-Hermitian sum of squares (moment-HSOS) hierarchy for complex polynomial optimization problems, where the sparsity includes correlative sparsity and term sparsity. We compare the strengths of the sparsity-adapted complex moment-HSOS hierarchy with the sparsity-adapted real moment-SOS hierarchy on either randomly generated complex polynomial optimization problems or the AC optimal power flow problem. The results of numerical experiments show that the sparsity-adapted complex moment-HSOS hierarchy provides a trade-off between the computational cost and the quality of obtained bounds for large-scale complex polynomial optimization problems. [ABSTRACT FROM AUTHOR]

Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon's conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods. [ABSTRACT FROM AUTHOR]

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua–Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an A n analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for γ = 1 (the Schur or GUE case), containing a product of n + 1 Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials. [ABSTRACT FROM AUTHOR]

In this article, we will show that for every choice of real numbers a 1 and a 2 , the set (loci) of solutions of all polynomials of the type x 3 + a 2 x 2 + a 1 x + d , where d ∈ R , can be characterized in terms of hyperbolas. Furthermore, relations between such hyperbolas and Steiner ellipses (inellipses and circumellipses) associated with cubics will be pointed out. [ABSTRACT FROM AUTHOR]

For β > 1 a real algebraic integer (the base), the finite alphabets A ⊂ Z which realize the identity Q (β) = Per A (β) , where Per A (β) is the set of complex numbers which are (β , A) -eventually periodic representations, are investigated. Comparing with the greedy algorithm, minimal and natural alphabets are defined. The natural alphabets are shown to be correlated to the asymptotics of the Pierce numbers of the base β and Lehmer's problem. The notion of rewriting trail is introduced to construct intermediate alphabets associated with small polynomial values of the base. Consequences on the representations of neighbourhoods of the origin in Q (β) , generalizing Schmidt's theorem related to Pisot numbers, are investigated. Applications to Galois conjugation are given for convergent sequences of bases γ s : = γ n , m 1 , ... , m s such that γ s - 1 is the unique root in (0, 1) of an almost Newman polynomial of the type - 1 + x + x n + x m 1 + ⋯ + x m s , n ≥ 3 , s ≥ 1 , m 1 - n ≥ n - 1 , m q + 1 - m q ≥ n - 1 for all q ≥ 1 . For β > 1 a reciprocal algebraic integer close to one, the poles of modulus < 1 of the dynamical zeta function of the β -shift ζ β (z) are shown, under some assumptions, to be zeroes of the minimal polynomial of β . [ABSTRACT FROM AUTHOR]

If ∑ k = 0 ∞ c k g k x is a formal series of orthonormal polynomials gk(x) on the real line with positive coefficients ck, then its partial sums un(x) are associated with Jacobi-type pencils. Therefore, they possess a recurrence relation and special orthonormality conditions. The cases where gk(x) are Jacobi or Laguerre polynomials are of especial interest. For a suitable selection of parameters ck, the partial sums un(x) turn into Sobolev orthogonal polynomials with a (3 × 3) matrix measure. Moreover, by the subsequent choice of parameters, we get differential equations for un. In the last case, the polynomials un(x) are solutions of generalized eigenvalue problems both in x and in n. [ABSTRACT FROM AUTHOR]

Let P(z) be a polynomial of degree at most n. We consider an operator N, which carries a polynomial P(z) into N [ P ] (z) : = ∑ j = 0 m λ j ( nz 2 ) j P (j) (z) j ! , where λ 0 , λ 1 , ... , λ m are such that all the zeros of u (z) = ∑ j = 0 m n j λ j z j lie in the half plane | z | ≤ | z - n 2 |. In this paper, we estimate the minimum and maximum modulii of N[P(z)] on | z | = 1 with restrictions on the zeros of P(z) and thereby obtain compact generalizations of some well known polynomial inequalities. [ABSTRACT FROM AUTHOR]

We establish a Wiman–Valiron theory of a polynomial series based on the Askey–Wilson operator D q , where q ∈ (0 , 1) . For an entire function f of log-order smaller than 2, this theory includes (i) an estimate which shows that f behaves locally like a polynomial consisting of the terms near the maximal term of its Askey–Wilson series expansion, and (ii) an estimate of D q n f compared to f. We then apply this theory in studying the growth of entire solutions to difference equations involving the Askey–Wilson operator. [ABSTRACT FROM AUTHOR]

Combinatorial properties of zeons have been applied to graph enumeration problems, graph colorings, routing problems in communication networks, partition-dependent stochastic integrals, and Boolean satisfiability. Power series of elementary zeon functions are naturally reduced to finite sums by virtue of the nilpotent properties of zeons. Further, the zeon extension of any analytic complex function has zeon polynomial representations on associated equivalence classes of zeons. In this paper, zeros of polynomials over complex zeons are considered. Existing results for real zeon polynomials are extended to the complex case and new results are established. In particular, a fundamental theorem of zeon algebra is established for spectrally simple zeros of complex zeon polynomials, and an algorithm is presented that allows one to find spectrally simple zeros when they exist. As an application, inverses of zeon extensions of analytic functions are computed using polynomial methods. [ABSTRACT FROM AUTHOR]

Given any system of Laurent-type power series, a criterion for the uniqueness of polyorthogonal polynomials of first type associated with this system is stated and proved, and explicit determinant representations generalizing E. Schmidt's formula for these polynomials are obtained. The proved statements supplement well-known results of the theory of orthogonal and polyorthogonal polynomials. [ABSTRACT FROM AUTHOR]

We study the gl 1 | 1 supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation gl 1 | 1 Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the angular coordinates of closely spaced signal sources using a rectangular antenna array with an arbitrary number of rows and columns. This problem is solved using the superresolution method of the minimum polynomial, which was previously proposed for the one-dimensional case of the angular resolution of signal sources. Two approaches to using this method are proposed, namely, the pseudospectral and the root techniques. The computational complexity is analyzed. A simpler alternative method for developing the resolution polynomial is also proposed in the special case of a two-row (two-column) antenna array for the root approach. The simulation results show the high efficiency of the angular resolution of the signal sources using the proposed methods. [ABSTRACT FROM AUTHOR]