In this paper, we derive a formula for the pluricomplex Green function of the bidisk with two poles of equal weights. In 2017, Kosiński, Thomas, and Zwonek proved the Lempert function and the pluricomplex Green function are equal on the bidisk, and their description of Lempert function was pivotal in computing the formula for the pluricomplex Green function. We divide the bidisk into two open regions, where the formula is found explicitly on the first region, and the other region is the union of a family of hypersurfaces. On each hypersurface, the formula is explicit up to a unimodular constant that is the root of a sixth degree polynomial. This derived formula for the bidisk leads to an explicit formula for the Carathéodory metric on the symmetrized bidisk up to a fourth degree polynomial. In 2004, Agler and Young found a formula for Carathéodory metric for the symmetrized bidisk that involves a supremum over the unimodular constants. The formula derived in this paper matches Agler and Young's formula, but the unimodular constant is determined by a 4th degree polynomial instead of the before mentioned supremum.

Against the background of increasing academic workload in higher education, this study examines the impact of academic course hours on college students' objective physical health, measured by comprehensive physical fitness scores, and explores potential gender differences and nonlinear patterns. Using a three-year longitudinal panel dataset (N = 305, 915 person-years) from a 2020 cohort at Changsha University of Science and Technology, we applied individual fixed effects models, supported by Hausman tests, to estimate the net effect of academic workload. The dependent variable was students' comprehensive physical fitness score, while the key independent variable was total annual academic (non-PE) course hours. Physical education (PE) course hours were included as a control variable. To identify nonlinear relationships, polynomial, logarithmic, and square root functional forms were tested, and gender-stratified analyses were conducted. Academic workload significantly and negatively affected students' physical health, with each additional academic hour associated with a 0.012-point decrease in physical fitness scores (p < 0.001). In contrast, PE course hours had a significant positive effect, increasing scores by 0.088 points per hour (p < 0.001). Gender differences were evident: the negative impact of academic workload was stronger among male students (B = -0.013) than among females (B = -0.009), and the health benefits of PE were also greater for males. Furthermore, the relationship between academic workload and physical health exhibited a clear nonlinear pattern. Physical health improved at low workload levels (<547 h), declined at moderate to high levels (547-1,087 h), and showed a slight marginal rebound at very high levels (>1,087 h). Among all tested specifications, the cubic model provided the best fit based on AIC and BIC criteria. Academic workload exerts a significant, gender-differentiated, and nonlinear influence on college students' physical health. Universities should maintain academic demands within an optimal range and ensure a balanced allocation between academic and physical education coursework. Targeted interventions, particularly for male students, are recommended to mitigate the adverse health effects of excessive academic burden.

UDC 512.552 We explicitly describe each unit of a group algebra $Z_{p} S_{3}$ for each positive prime $p \geq 5$ by using a characterization of the group algebra of the metacyclic group $G= \langle x,c\colon x^{3}=1,\ c^{n}=1,\ cxc^{-1 } = x^{-1} \rangle$ over the finite field $F$ of characteristic $p,$ where $p$ is a positive prime such that $p \nmid 3n.$ Based on our findings, we pose a conjecture on the number of roots of some explicit polynomials over the prime field $\mathbb{Z}_{p}$ for further academic explorations.

The in–out degree polynomial of a directed graph is a binary polynomial that combines the indegree polynomial and the outdegree polynomial. In this paper, we give the in–out degree polynomials and their generating function of the directed Fibonacci cubes. As a sequence, we obtain the indegree polynomials and the outdegree polynomials of the cubes. We also study some properties of the coefficients and roots of these univariate polynomials. Finally, we find a close relationship between the indegree polynomials of the cubes and the Morgan–Voyce polynomials, as well as the Fibonacci polynomials.

Abstract In this article, the large-time asymptotic wave dynamics of rogue curves are analytically investigated and numerically confirmed in the Davey–Stewartson (DS) I equation. We show that, when time in bilinear expressions of the rogue curves is large, a certain number of localized lump-shaped waves would arise on the uniform background, exhibiting various wave patterns. We further show that, as time increases, the individual lump-shaped wave asymptotically evolves into a line soliton on the constant background that persist at large time. By performing large-time asymptotic analysis, we reveal that such wave patterns as well as the numbers of lump-shaped waves can be analytically determined by the structure of nonzero roots of the Wronskian-Hermite polynomials. Our asymptotic predictions are compared to true solutions quantitatively and excellent agreement is obtained.

A classic exercise in Galois theory asks to show that if p is a prime number and f ( X ) is an irreducible rational polynomial of degree p with exactly p − 2 real roots, and if L is the splitting field of f ( X ) over Q , then Gal ( L / Q ) ≅ S p , the symmetric group of degree p. It is therefore of interest to construct such polynomials. We present such a construction. More precisely, let n ≥ 1 be an integer. Given a set of n different values { z 1 , … , z n } in C that is stable under complex conjugation, we construct in an effective way irreducible polynomials in Q [ X ] with integral coefficients of degree n (respectively monic and integral of degree n + 1 ) with roots α 1 , … , α n (respectively, roots α 0 , α 1 , … , α n ) such that to any given real number ϵ > 0 one has | z k − α k | < ϵ for 1 ≤ k ≤ n , thus also recovering a result by Theodore S. Motzkin from 1947 using a new approach by invoking Rouché’s theorem from complex analysis.

We propose a novel method to robustly and efficiently compute the maximum allowable step sizes so that the 3D high-order finite elements continuously preserve geometric validity when moving along the given directions with positive step sizes smaller than the computed ones. We transform the problem of finding the maximum allowable step sizes to one of solving roots of cubic polynomials. To use interval arithmetic to avoid numerical issues in cubic equation solving, we completely enumerate the roots of cubic polynomials and apply the interval version of the Newton-Raphson iteration. The effectiveness of our algorithm is demonstrated through extensive testing. Compared to the state-of-the-art method, our algorithm achieves higher efficiency.

DOA Estimation by Polynomial Rooting for Electronically Scanned CRLH Leaky-Wave Antennas

Numerical homotopy continuation methods are known to be accurate and fast for obtaining roots of univariate polynomials with random coefficients. Due to a result of Kac (1943), which was extended by Edelman and Kostlan (1995), we know that the roots of such polynomials tend to be uniformly distributed on the unit circle, and due to the low condition numbers of such roots, offer a "best case" scenario for testing numerical root-finding algorithms. This paper considers the accuracy and computation cost of homotopy methods of average case polynomials such as the low degree Mandelbrot polynomials, and polynomials generated from random roots. For a worst case polynomial, we look at the Wilkinson polynomial with all positive roots. We take a novel approach in studying both numerical pseudozeros of the target polynomial, and the exact pseudozeros given by the homotopy. We confirm the practitioner's expectation that accuracy of high-speed homotopy methods are highly dependent on how well the start system is scaled to fit the target roots. Thus, the so-called Bézout start system used to find roots on the unit circle is nearly ideal. We show how to adapt these insights to work with other polynomials, including changing from the monomial basis to the Lagrange basis.

Although the Durand-Kerner method is widely used across various fields of computer science, especially in numerical computing, it continues to encounter challenges in locating roots of high-degree polynomials, such as issues with accuracies of roots of the polynomial zeros. Our initial tests and observations on several methods for finding polynomial roots revealed that the roots' accuracy starts to degrade noticeably for polynomials where the degree exceeds 10. Based on considerations of algebraic concepts involving polynomial vector spaces, we introduce an improvement of the Durand-Kerner algorithm aimed at improving root precision. This approach includes targeted refinements in coefficient evaluation, identification of root types, and iterative polishing techniques. We also conducted a comparative evaluation to assess its effectiveness against the original Durand Kerner method and MATLAB's roots() function. Overall, the enhanced algorithm delivers superior accuracy for complex roots—particularly in cases involving multiple zero or integer roots—outperforming both benchmarks, but its execution time increases substantially with polynomial degree.

Let θ be a root of a monic polynomial h(x)∈Z[x] of degree n≥2. We say that h(x) is monogenic if it is irreducible over Q and {1,θ,θ2,…,θn−1} is a basis for the ring ZK of integers of K=Q(θ). We investigate monogenity of number fields generated by roots of compositions of two binomials. We characterise all the primes dividing the index of the subgroup Z[θ] in ZK where K=Q(θ) with θ having minimal polynomial F(x)=(xm−b)n−a∈Z[x], m≥1 and n≥2. As an application, we provide a class of pairs of binomials f(x)=xn−a and g(x)=xm−b having the property that both f(x) and f(g(x)) are monogenic.

Deformation of polynomials is a kind of operation where we add a new variable to the original polynomial. In our case, suppose P is a monic polynomial of degree n with complex coefficients. We evolve P with respect to time by heat flow, creating a function P(t,z) of two variables with given initial dataP(0,z)=P(z)for whichtP(t,z)=zzP(t,z). In this paper, we focus on the deformed polynomial P(t,z). First, we proved the Taylor series representation of deformed polynomial. Then we apply the results to the classical Hermite polynomials and extend to the case of matrix-valued polynomials. From the inspiration of deformed polynomials roots movement, we proved the behavior of Hermite polynomials after heat flow deformation and got an explicit formula. For further work, similar to what we have done in this paper, we want to have an explicit formula for deformed matrix Hermite polynomials and give a proof.

The article presents the method for analyzing the stability of linear matrix differential equations with constant coefficients. One of the classical typesof such equations is the class of linear matrix differential equations, which includes the Lyapunov equation as a particular case. Matrix differentialequations arise in problems of stability theory, practical stability, optimal control theory, and state estimation of systems under uncertainty. Therefore,it is necessary to compute and analyze the qualitative properties of solutions to matrix differential equations. This involves addressing problems of existence, uniqueness, continuation, and analysis of stability conditions for various types of such mathematical equations. The method proposed in thearticle is based on algebraic properties of eigenvalues, Jordan forms of matrices, and characteristics of polynomial roots. A theorem is established regardingthe conditions for stability, asymptotic stability, and instability of solutions to linear matrix differential equations with constant coefficients.The developed approach includes the computation of the maximal real parts of eigenvalues and the analysis of the Jordan form structure of the systemmatrices. As a consequence, corresponding stability conditions for the Lyapunov matrix equation are also obtained. An algorithm is proposed for computingthe maximal real part of the roots of a polynomial, as well as for finding all roots. The approach relies on the Routh – Hurwitz theorem. The article also presents results of computational experiments.

Tricomplex numbers are a generalization of bicomplex numbers. In this paper, we detail a technique for finding the roots of tricomplex polynomials. We generalized then the process to multicomplex polynomials. We first give an idempotent-representation of tricomplex numbers and reduce the working method to complex polynomials. We give examples to illustrate the different situations. Finally, for a multicomplex polynomial, we explain a reduction process ending to search roots in the complex field. Combining these give the roots for multicomplex polynomials.

We prove that if a polynomial with rational coefficients has a root mod p for every large prime p, then it has a real root. We show with examples how to use Chebotarev’s density theorem to study roots of polynomials mod p, leading up to our proof. As an application, we show that the primes can’t be covered by finitely many positive definite binary quadratic forms.

This paper presents a method to accurately determine the reachable set (RS) of spacecraft with deterministic state after a single velocity impulse with an arbitrary direction, which is appropriate for the RS in both the state and observation spaces under arbitrary dynamics, extending the applications of current single-impulse RS methods from two-body to arbitrary dynamics. First, the single-impulse RS model is generalized as a family of two-variable parameterized polynomials in the differential algebra scheme. Then, using the envelope theory, the RS boundary is identified by solving the envelope equation. A framework is proposed to reduce the complexity of solving the envelope equation by converting it to the problem of searching for the root of a one-variable polynomial. Moreover, a high-order local polynomial approximation for the RS envelope is derived to improve computational efficiency. The method successfully determines the RSs of two near-rectilinear halo orbits in the cislunar space. Simulation results show that the RSs in both state and observation spaces can be accurately approximated under the three-body dynamics, with relative errors of less than 0.0658%. In addition, using the local polynomial approximation, the computational time for solving the envelope equation is reduced by more than 84%.

ABSTRACT This article presents an algebraic and inductive proof of Descartes’ Rule of Signs, dispensing with the use of Analysis tools. This article presents an algebraic and inductive proof of Descartes’ Rule of Signs, avoiding the use of Analysis tools such as Rolle’s Theorem or the Intermediate Value Theorem. The theorem establishes an upper bound on the number of positive (or negative) real roots of a real polynomial based on the number of sign variations of its coefficients. The novelty of this proof lies in its purely algebraic and constructive nature, structured around elementary lemmas on signs and induction on the number of positive roots, which gives it educational value for secondary and lower secondary education. We discuss the history of the theorem, various existing proofs, and indicate how the topic can be explored pedagogically in the classroom. Keywords: Algebraic proof, Descartes’ Theorem, lemmas, induction, didactic value, High School.

The resolvent degree \operatorname{rd}_{\mathbb{C}}(n) is the smallest integer d such that a root of the general polynomial f(x) = x^{n} + a_{1} x^{n-1}+ \dots + a_{n} can be expressed as a composition of algebraic functions in at most d variables with complex coefficients. It is known that \operatorname{rd}_{\mathbb{C}}(n)=1 when n\leqslant 5 . Hilbert was particularly interested in the next three cases: he asked if \operatorname{rd}_{\mathbb{C}}(6)=2 (Hilbert’s Sextic conjecture), \operatorname{rd}_{\mathbb{C}}(7)=3 (Hilbert’s 13th problem) and \operatorname{rd}_{\mathbb{C}}(8)=4 (Hilbert’s Octic conjecture). These problems remain open. It is known that \operatorname{rd}_{\mathbb{C}}(6)\leqslant 2 , \operatorname{rd}_{\mathbb{C}}(7)\leqslant 3 and \operatorname{rd}_{\mathbb{C}}(8)\leqslant 4 . It is not known whether or not \operatorname{rd}_{\mathbb{C}}(n) can be &gt;1 for any n\geqslant 6 .In this paper, we show that all three of Hilbert’s conjectures can fail if we replace \mathbb{C} with a base field of positive characteristic.

Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. We determine the dimension, degree, singular locus and defining equations of these varieties. We explain how they play the role of toric varieties in sparse polynomial root finding, when monomials are replaced by Chebyshev polynomials. We present numerical root finding algorithms that exploit our results.

The quadratic sieve method is a core tool in number theory. In this paper, we present two optimization methods for the Quadratic Sieve algorithm. In the sieving process, the original polynomial root value accumulation step is changed from the original 𝑑𝑖 to the 𝑚𝑑𝑖 (m is a small integer), which can change the complexity from 𝑂(𝑛) to 𝑂 ( 𝑛 𝑚 ). Another optimization method is that for all parameter lookup steps, the original traversal lookup can be changed to an efficient binary search, which can change the complexity of the loop from 𝑂(𝑛) to 𝑂(𝑙𝑜𝑔𝑛) . This enhancement reduces the computational complexity of RSA modulus factorization in practical settings.