An algorithm for finding low degree rational solutions to the Schur coefficient problem

TL;DR

This paper introduces an algorithm to find all rational functions with specified Taylor coefficients, bounded supremum norm, and degree constraints, extending to cases where degree is less than the number of coefficients.

Contribution

The paper presents a novel algorithm for constructing rational solutions with prescribed Taylor coefficients and degree bounds, including cases where degree is less than the number of coefficients.

Findings

Algorithm produces all such rational functions for given constraints.

Extends to cases with degree less than the number of Taylor coefficients.

Provides a systematic method for solving the Schur coefficient problem.

Abstract

We present an algorithm producing all rational functions $f$ with prescribed $n+1$ Taylor coefficients at the origin and such that $\|f\|_\infty\le 1$ and $\deg f\le k$ for every fixed $k\ge n$. The case where $k<n$ is also discussed.