Related papers: Zeros of replicable functions

Given a finite index subgroup $\Gamma$ of ${\rm{PSL}}_2(\Bbb{Z})$, we investigate Belyi functions on the corresponding modular curve $X(\Gamma)$ by introducing two methods for constructing such functions. Numerous examples have been worked…

We produce canonical sets of right coset representatives for the congruence subgroups $\Gamma_0(N)$, $\Gamma_1(N)$ and $\Gamma(N)$, and prove that the corresponding fundamental domains are connected. Key to our construction is a study of…

We give bounds on the degree of generators for the ideal of relations of the graded algebras of modular forms with coefficients in $\mathbb{Q}$ over congruence subgroups $\Gamma_0(N)$ for $N$ satisfying some congruence conditions and for…

From a sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ whose coefficients are numbers in this sequence. We…

There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic…

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum possible modulus of occurring zeros via a…

Let $\Gamma$ be a finite group acting linearly on $\C^n$, freely outside the origin, and let $N$ be the number of conjugacy classes of $\Gamma$ minus one. A construction of Kronheimer of moduli spaces $X_\zeta$ of translation-invariant…

The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $\zeta_{\beta}(z)$ of the $\beta$-shift, for $\beta > 1$ close to one, is investigated. We briefly recall how this family is…

For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(\alpha_n, \beta_n)} (x), L_n^{(\alpha_n)} (x),$\break $H_n^{(\gamma_n)} (x)$ the limit distributions of the zeros are found, when the sequences $\alpha_n$ or $\beta_n$ tend…

Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in $\mathbb{R}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma\subset \mathbb{R}\mathrm{P}^n$ such…

We introduce and analyse a general class of not necessarily bounded multiplicative functions, examples of which include the function $n \mapsto \delta^{\omega (n)}$, where $\delta \neq 0$ and where $\omega$ counts the number of distinct…

Given a well-ordered semi-group $\Gamma$ with a minimal system of generators of ordinal type at most $\omega n$ and of rational rank $r$, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation…

Locating the zeros of quaternionic polynomials is a fundamental problem with significant implications across scientific and engineering disciplines, yet the noncommutative nature of quaternion multiplication makes it fundamentally more…

We consider Laguerre polynomials $L_n^{(\alpha_n)}(nz)$ with varying negative parameters $\alpha_n$, such that the limit $A = -\lim_n \alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an…

Let $\Gamma \subseteq \text{SL}_2(\mathbb{R})$ be a genus zero Fuchsian group of the first kind having $\infty$ as a cusp, and let $E_{2 k}^{\Gamma}$ be the holomorphic Eisenstein series associated with $\Gamma$ for the $\infty$ cusp that…

Let $p(z)=z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\ldots+a_1z+a_0$ be a complex polynomial with $a_0\neq 0$ and $n\geq 3$. Several new upper bounds for the moduli of the zeros of $p$ are developed. In particular, if…

Laguerre's theorem regarding the number of non-real zeros of a polynomial and its image under certain linear operators is generalized. This generalization is then used to (1) exhibit a number of previously undiscovered complex zero…

We use recent results about linking the number of zeros on algebraic varieties over $\mathbb{C}$, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products…

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an…

In this paper, we study the zero sets of the confluent hypergeometric function $_{1}F_{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!(\gamma)_{n}}z^{n}$, where $\alpha, \gamma, \gamma-\alpha\not\in \mathbb{Z}_{\leq 0}$, and…