The winding number of PF+1 for polynomials P and meromorphic extendibility of F

TL;DR

This paper investigates a conjecture relating the winding number of Pf+1 for polynomials P to the holomorphic extendibility of functions on the unit disk, proving a related meromorphic extendibility result for smooth functions.

Contribution

It establishes a meromorphic extendibility criterion for smooth functions with finitely many zeros, supporting the conjecture for real analytic functions.

Findings

Proves a meromorphic extension result for smooth functions with finitely many zeros.

Supports the conjecture for real analytic functions.

Provides a new perspective on the winding number condition and extendibility.

Abstract

Let D be the open unit disc in C. The paper deals with the following conjecture: If f is a continuous function on bD such that the change of argument of Pf+1 around bD is nonnegative for every polynomial P such that Pf+1 has no zero on bD then f extends holomorphically through D. We prove a related result on meromorphic extendibility for smooth functions with finitely many zeros of finite order, which, in particular, implies that the conjecture holds for real analytic functions.