Pfaffian control of some polynomials involving the $j$--function and Weierstrass elliptic functions

TL;DR

This paper establishes new bounds on the zeros of polynomials involving the $j$-function and Weierstrass elliptic functions with rectangular lattices, based on their boundary behavior within fundamental domains.

Contribution

It introduces novel bounds on zeros of these special functions, linking their interior zeros to boundary properties, advancing understanding in complex analysis and elliptic functions.

Findings

Bounds on zeros of polynomials with $j$-function and elliptic functions

Control of zeros via boundary behavior on fundamental domains

Enhanced understanding of elliptic function zeros

Abstract

We give new bounds on the zeros of polynomials in $z$ and the $j$--function, and $z$ and Weierstrass elliptic functions with rectangular associated lattice, controlling the zeros of these functions by their tame behaviour on the boundaries of their respective fundamental domains.