Power Series Approximations to Fekete Polynomials

TL;DR

This paper investigates the approximation of Fekete polynomials, defined by Legendre symbol coefficients, using power series of algebraic functions, and derives explicit recurrence relations for their coefficients.

Contribution

It introduces methods to approximate Fekete polynomials with algebraic power series and derives explicit recurrence relations for their coefficients.

Findings

Power series can approximate Fekete polynomials effectively.

Explicit recurrence relations for algebraic function coefficients are obtained.

Results enhance understanding of the algebraic structure of Fekete polynomials.

Abstract

We study how well Fekete polynomials $$ F_p(X) = \sum_{n=0}^{p-1} \left(\frac{n}{p}\right) X^n \in {\mathbb Z}[X] $$ with the coefficients given by Legendre symbols modulo a prime $p$, can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions.