相关论文: Lemniscates as Trajectories of Quadratic Different…
In this note, we study the fingerprints of closed smooth curves that are trajectories of a particular Strebel quadratic differentials. It is a generalization of the cases of polynomials and rational lemniscates.
We discuss some aspects of the theory of recognition of two-dimensional shapes by means of fingerprints of Jordan curves. An interesting approach to problems on shape recognition suggested by P.~Ebenfelt, D.~Khavinson, and H.~Shapiro and…
It is known that the exponential transform of a quadrature domain is a rational function for which the denominator has a certain separable form. In the present paper we show that the exponential transform of lemniscate domains in general…
We prove that lemniscates (i.e., sets of the form $|P(z)|=1$ where $P$ is a complex polynomial) are irreducible real algebraic curves.
Cubic and quartic non-autonomous differential equations with continuous piecewise linear coefficients are considered. The main concern is to find the maximum possible multiplicity of periodic solutions. For many classes, we show that the…
A lemniscate of a complex polynomial $Q_n$ of degree $n$ is a sublevel set of its modulus, i.e., of the form $\{z \in \mathbb{C}: |Q_n(z)| < t\}$ for some $t>0.$ In general, the number of connected components of this lemniscate can vary…
Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic…
This article studies the Galois groups that arise from division points of the lemniscate. We compute these Galois groups two ways: first, by class field theory, and second, by proving the irreducibility of lemnatomic polynomials, which are…
A novel basis of discrete analytic polynomials on a rhombic lattice is introduced and the associated convolution product is studied. A class of discrete analytic functions that are rational with respect to this product is also described.
We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…
We consider families of polynomial lemniscates in the complex plane and determine if they bound a Jordan domain. This allows us to find examples of regions for which we can calculate the projection of $\bar{z}$ to the Bergman space of the…
A rational lemniscate is a level set of $|r|$ where $r: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This…
Given a generic rational curve $C$ in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly $C$. Our construction is based on the factorization of polynomials over dual…
We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them /periodic/ and…
In this paper, we consider linear differential equations satisfied by the generating function for Hermite polynomials and derive some new identities involving those polynomials.
The trajectories of a rational motion given by a polynomial of degree n in the dual quaternion model of rigid body displacements are generically of degree 2n. In this article we study those exceptional motions whose trajectory degree is…
This paper is devoted to several new results concerning (standard) octonion polynomials. The first is the determination of the roots of all right scalar multiples of octonion polynomials. The roots of left multiples are also discussed,…
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined…
We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and…
In a previous work, both the constants of motion of a classical system and the symmetries of the corresponding quantum version have been computed with the help of factorizations. As their expressions were not polynomial, in this paper the…