Gaps in Taylor series of algebraic functions

TL;DR

This paper investigates the zeros in the Taylor series coefficients of rational functions on algebraic curves, providing bounds on their frequency of vanishing.

Contribution

It introduces an upper bound on how often the Taylor series coefficients of algebraic functions can be zero, advancing understanding of their local expansions.

Findings

Established an upper bound on zero coefficients in Taylor series

Analyzed the behavior of Taylor series of rational functions on algebraic curves

Contributed to the theory of algebraic function expansions

Abstract

Let $f$ be a rational function on an algebraic curve over the complex numbers. For a point $p$ and local parameter $x$ we can consider the Taylor series for $f$ in the variable $x$. In this paper we give an upper bound on the frequency with which the terms in the Taylor series have $0$ as their coefficient.