Related papers: On the number of zeros of certain rational harmoni…

An important theorem of Khavinson & Neumann (Proc. Amer. Math. Soc. 134(4), 2006) states that the complex harmonic function $r(z) - \bar{z}$, where $r$ is a rational function of degree $n \geq 2$, has at most $5 (n - 1)$ zeros. In this note…

General Relativity gives that finitely many point masses between an observer and a light source create many images of the light source. Positions of these images are solutions of $r(z)=\bar{z},$ where $r(z)$ is a rational function. We study…

We study how adding certain poles to rational harmonic functions of the form $R(z)-\bar{z}$, with $R(z)$ rational and of degree $d\geq 2$, affects the number of zeros of the resulting functions. Our results are motivated by and generalize a…

We study the effect of constant shifts on the zeros of rational harmomic functions $f(z) = r(z) - \conj{z}$. In particular, we characterize how shifting through the caustics of $f$ changes the number of zeros and their respective…

Generalizing several previous results in the literature on rational harmonic functions, we derive bounds on the maximum number of zeros of functions $f(z) = \frac{p(z)}{q(z)} - \overline{z}$, which depend on both $\mathrm{deg}(p)$ and…

It has been conjectured (astro-ph/0103463) that a gravitational lens consisting of n point masses can not produce more than 5(n-1) images as is known to be the case for n = 2 and 3. The reasoning is based on the number of finite limit…

We investigate the zeros of two one-parameter families of harmonic functions and describe how the number of zeros depends on the parameter. Our functions have the property that all zeros lie on certain rays in the complex plane and thus we…

D. Khavinson and G. Swiatek proved that harmonic polynomials p(z)+q(z), where p is holomorphic, q is antiholomorphic, and deg p = n > 1 = deg q, can have at most 3n-2 complex zeros. We show that this bound is sharp for all n by proving a…

We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of…

A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…

We discuss gravitational lensing by elliptical galaxies with some particular mass distributions. Using simple techniques from the theory of quadrature domains and the Schwarz function (cf. \cite{Sh}) we show that when the mass density is…

The equation in the title describes the number of bright images of a point source under lensing by an elliptic object with isothermal density. We prove that this equation has at most 6 solutions. Any number of solutions from 1 to 6 can…

In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics…

We prove the existence of complex polynomials $p(z)$ of degree $n$ and $q(z)$ of degree $m<n$ such that the harmonic polynomial $ p(z) + \overline{q(z)}$ has at least $\lceil n \sqrt{m} \rceil$ many zeros. This provides an array of new…

We consider gravitational lensing of a background source by a finite system of point-masses. The problem of determining the maximum possible number of lensed images has been completely resolved in the single-plane setting (where the point…

Let $\Re_n$ be the set of all rational functions of the type $r(z) = p(z)/w(z),$ where $p(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-a_j)$, $|a_j|>1$ for $1\leq j\leq n$. In this paper, we set up some results…

The valence of a function f at a point $z_0$ is the number of distinct, finite solutions to $f(z) = z_0.$ In this paper, we bound the valence of complex-valued harmonic polynomials in the plane for some special harmonic polynomials of the…

It is shown that the lens equation for a binary gravitational lens being a set of two coupled real fifth-order algebraic equations (equivalent to a single complex equation of the same order) can be reduced to a single real fifth-order…

Let $r(z)$ be a rational function with at most $n$ poles, $a_1, a_2, \ldots, a_n,$ where $|a_j| > 1,$ $1\leq j\leq n.$ This paper investigates the estimate of the modulus of the derivative of a rational function $r(z)$ on the unit circle.…

We consider multivariate polynomials and investigate how many zeros of multiplicity at least $r$ they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of…